id stringlengths 11 15 | title stringlengths 7 171 | problem stringlengths 9 4.33k | solution stringlengths 6 19k | problem_wikitext stringlengths 9 4.42k | solution_wikitext stringlengths 7 19.1k | proof_title stringlengths 9 171 | theorem_url stringlengths 34 198 | proof_url stringlengths 36 198 | categories listlengths 0 9 | theorem_references listlengths 0 36 | proof_references listlengths 0 253 |
|---|---|---|---|---|---|---|---|---|---|---|---|
proofwiki-1400 | Dedekind's Theorem | Let $\tuple {L, R}$ be a Dedekind cut of the set of real numbers $\R$.
Then there exists a unique real number which is a producer of $\tuple {L, R}$. | === Proof of Uniqueness ===
{{AimForCont}} both $\alpha$ and $\beta$ produce $\tuple {L, R}$.
By the Trichotomy Law for Real Numbers either $\beta < \alpha$ or $\alpha < \beta$.
Suppose that $\beta < \alpha$.
From Real Numbers are Densely Ordered, there exists at least one real number $c$ such that $\beta < c$ and $c <... | Let $\tuple {L, R}$ be a [[Definition:Dedekind Cut|Dedekind cut]] of the set of [[Definition:Real Number|real numbers]] $\R$.
Then there exists a [[Definition:Unique|unique]] [[Definition:Real Number|real number]] which is a [[Definition:Producer of Dedekind Cut|producer]] of $\tuple {L, R}$. | === Proof of Uniqueness ===
{{AimForCont}} both $\alpha$ and $\beta$ [[Definition:Producer of Dedekind Cut|produce]] $\tuple {L, R}$.
By the [[Trichotomy Law for Real Numbers]] either $\beta < \alpha$ or $\alpha < \beta$.
Suppose that $\beta < \alpha$.
From [[Real Numbers are Densely Ordered]], there exists at leas... | Dedekind's Theorem/Proof 3 | https://proofwiki.org/wiki/Dedekind's_Theorem | https://proofwiki.org/wiki/Dedekind's_Theorem/Proof_3 | [
"Real Analysis",
"Dedekind Cuts",
"Dedekind's Theorem"
] | [
"Definition:Dedekind Cut",
"Definition:Real Number",
"Definition:Unique",
"Definition:Real Number",
"Definition:Producer of Dedekind Cut"
] | [
"Definition:Producer of Dedekind Cut",
"Trichotomy Law for Real Numbers",
"Real Numbers are Densely Ordered",
"Definition:Real Number",
"Definition:Dedekind Cut",
"Definition:Set Partition",
"Definition:Disjoint Sets",
"Definition:Contradiction",
"Definition:Contradiction",
"Definition:Unique",
... |
proofwiki-1401 | Complex Addition is Closed | The set of complex numbers $\C$ is closed under addition:
:$\forall z, w \in \C: z + w \in \C$ | From the informal definition of complex numbers, we define the following:
:$z = x_1 + i y_1$
:$w = x_2 + i y_2$
where $i = \sqrt {-1}$ and $x_1, x_2, y_1, y_2 \in \R$.
Then from the definition of complex addition:
:$z + w = \paren {x_1 + x_2} + i \paren {y_1 + y_2}$
From Real Numbers under Addition form Group, real add... | The [[Definition:Set|set]] of [[Definition:Complex Number|complex numbers]] $\C$ is [[Definition:Closed Algebraic Structure|closed]] under [[Definition:Complex Addition|addition]]:
:$\forall z, w \in \C: z + w \in \C$ | From the informal definition of [[Definition:Complex Number/Definition 1|complex numbers]], we define the following:
:$z = x_1 + i y_1$
:$w = x_2 + i y_2$
where $i = \sqrt {-1}$ and $x_1, x_2, y_1, y_2 \in \R$.
Then from the definition of [[Definition:Complex Addition|complex addition]]:
:$z + w = \paren {x_1 + x_2}... | Complex Addition is Closed/Proof 1 | https://proofwiki.org/wiki/Complex_Addition_is_Closed | https://proofwiki.org/wiki/Complex_Addition_is_Closed/Proof_1 | [
"Complex Addition is Closed",
"Complex Addition",
"Algebraic Closure"
] | [
"Definition:Set",
"Definition:Complex Number",
"Definition:Closure (Abstract Algebra)/Algebraic Structure",
"Definition:Addition/Complex Numbers"
] | [
"Definition:Complex Number/Definition 1",
"Definition:Addition/Complex Numbers",
"Real Numbers under Addition form Group",
"Definition:Addition/Real Numbers",
"Definition:Closure (Abstract Algebra)/Algebraic Structure"
] |
proofwiki-1402 | Complex Addition is Closed | The set of complex numbers $\C$ is closed under addition:
:$\forall z, w \in \C: z + w \in \C$ | From the formal definition of complex numbers, we have:
:$z = \tuple {x_1, y_1}$
:$w = \tuple {x_2, y_2}$
where $x_1, x_2, y_1, y_2 \in \R$.
Then from the definition of complex addition:
:$z + w = \tuple {x_1 + x_2, y_1 + y_2}$
From Real Numbers under Addition form Group, real addition is closed.
So:
:$\paren {x_1 + x_... | The [[Definition:Set|set]] of [[Definition:Complex Number|complex numbers]] $\C$ is [[Definition:Closed Algebraic Structure|closed]] under [[Definition:Complex Addition|addition]]:
:$\forall z, w \in \C: z + w \in \C$ | From the formal definition of [[Definition:Complex Number/Definition 2|complex numbers]], we have:
:$z = \tuple {x_1, y_1}$
:$w = \tuple {x_2, y_2}$
where $x_1, x_2, y_1, y_2 \in \R$.
Then from the definition of [[Definition:Complex Number/Definition 2/Addition|complex addition]]:
:$z + w = \tuple {x_1 + x_2, y_1 + ... | Complex Addition is Closed/Proof 2 | https://proofwiki.org/wiki/Complex_Addition_is_Closed | https://proofwiki.org/wiki/Complex_Addition_is_Closed/Proof_2 | [
"Complex Addition is Closed",
"Complex Addition",
"Algebraic Closure"
] | [
"Definition:Set",
"Definition:Complex Number",
"Definition:Closure (Abstract Algebra)/Algebraic Structure",
"Definition:Addition/Complex Numbers"
] | [
"Definition:Complex Number/Definition 2",
"Definition:Complex Number/Definition 2/Addition",
"Real Numbers under Addition form Group",
"Definition:Addition/Real Numbers",
"Definition:Closure (Abstract Algebra)/Algebraic Structure"
] |
proofwiki-1403 | Complex Addition is Associative | The operation of addition on the set of complex numbers $\C$ is associative:
:$\forall z_1, z_2, z_3 \in \C: z_1 + \paren {z_2 + z_3} = \paren {z_1 + z_2} + z_3$ | From the definition of complex numbers, we define the following:
{{begin-eqn}}
{{eqn | l = z_1
| o = :=
| r = \tuple {x_1, y_1}
}}
{{eqn | l = z_2
| o = :=
| r = \tuple {x_2, y_2}
}}
{{eqn | l = z_3
| o = :=
| r = \tuple {x_3, y_3}
}}
{{end-eqn}}
where $x_1, x_2, x_3, y_1, y_2, y_3 \... | The operation of [[Definition:Complex Addition|addition]] on the [[Definition:Set|set]] of [[Definition:Complex Number|complex numbers]] $\C$ is [[Definition:Associative Operation|associative]]:
:$\forall z_1, z_2, z_3 \in \C: z_1 + \paren {z_2 + z_3} = \paren {z_1 + z_2} + z_3$ | From the definition of [[Definition:Complex Number/Definition 2|complex numbers]], we define the following:
{{begin-eqn}}
{{eqn | l = z_1
| o = :=
| r = \tuple {x_1, y_1}
}}
{{eqn | l = z_2
| o = :=
| r = \tuple {x_2, y_2}
}}
{{eqn | l = z_3
| o = :=
| r = \tuple {x_3, y_3}
}}
{{end... | Complex Addition is Associative | https://proofwiki.org/wiki/Complex_Addition_is_Associative | https://proofwiki.org/wiki/Complex_Addition_is_Associative | [
"Complex Addition is Associative",
"Complex Addition",
"Associative Law of Addition"
] | [
"Definition:Addition/Complex Numbers",
"Definition:Set",
"Definition:Complex Number",
"Definition:Associative Operation"
] | [
"Definition:Complex Number/Definition 2",
"Real Addition is Associative"
] |
proofwiki-1404 | Complex Addition is Commutative | The operation of addition on the set of complex numbers is commutative:
:$\forall z, w \in \C: z + w = w + z$ | From the definition of complex numbers, we define the following:
{{begin-eqn}}
{{eqn | l = z
| o = :=
| r = \tuple {x_1, y_1}
}}
{{eqn | l = w
| o = :=
| r = \tuple {x_2, y_2}
}}
{{end-eqn}}
where $x_1, x_2, y_1, y_2 \in \R$.
Then:
{{begin-eqn}}
{{eqn | l = z + w
| r = \tuple {x_1, y_1} + ... | The operation of [[Definition:Complex Addition|addition]] on the [[Definition:Set|set]] of [[Definition:Complex Number|complex numbers]] is [[Definition:Commutative Operation|commutative]]:
:$\forall z, w \in \C: z + w = w + z$ | From the definition of [[Definition:Complex Number/Definition 2|complex numbers]], we define the following:
{{begin-eqn}}
{{eqn | l = z
| o = :=
| r = \tuple {x_1, y_1}
}}
{{eqn | l = w
| o = :=
| r = \tuple {x_2, y_2}
}}
{{end-eqn}}
where $x_1, x_2, y_1, y_2 \in \R$.
Then:
{{begin-eqn}}
{{e... | Complex Addition is Commutative | https://proofwiki.org/wiki/Complex_Addition_is_Commutative | https://proofwiki.org/wiki/Complex_Addition_is_Commutative | [
"Complex Addition",
"Commutative Law of Addition",
"Examples of Commutative Operations"
] | [
"Definition:Addition/Complex Numbers",
"Definition:Set",
"Definition:Complex Number",
"Definition:Commutative/Operation"
] | [
"Definition:Complex Number/Definition 2",
"Real Addition is Commutative"
] |
proofwiki-1405 | Integers are Countably Infinite | The set $\Z$ of integers is countably infinite. | Define the inclusion mapping $i: \N \to \Z$.
From Inclusion Mapping is Injection, $i: \N \to \Z$ is an injection.
Thus there exists an injection from $\N$ to $\Z$.
Hence $\Z$ is infinite.
Next, let us arrange $\Z$ in the following order:
:$\Z = \set {0, 1, -1, 2, -2, 3, -3, \ldots}$
Then we can directly see that we can... | The [[Definition:Set|set]] $\Z$ of [[Definition:Integer|integers]] is [[Definition:Countably Infinite Set|countably infinite]]. | Define the [[Definition:Inclusion Mapping|inclusion mapping]] $i: \N \to \Z$.
From [[Inclusion Mapping is Injection]], $i: \N \to \Z$ is an [[Definition:Injection|injection]].
Thus there exists an injection from $\N$ to $\Z$.
Hence $\Z$ is [[Definition:Infinite|infinite]].
Next, let us arrange $\Z$ in the followin... | Integers are Countably Infinite | https://proofwiki.org/wiki/Integers_are_Countably_Infinite | https://proofwiki.org/wiki/Integers_are_Countably_Infinite | [
"Integers",
"Countable Sets"
] | [
"Definition:Set",
"Definition:Integer",
"Definition:Countably Infinite/Set"
] | [
"Definition:Inclusion Mapping",
"Inclusion Mapping is Injection",
"Definition:Injection",
"Definition:Infinite",
"Definition:Mapping",
"Definition:Injection",
"Definition:Injection",
"Domain of Injection to Countable Set is Countable"
] |
proofwiki-1406 | Permutation of Determinant Indices | Let $\mathbf A = \sqbrk a_n$ be a square matrix of order $n$ over a field.
Let $\lambda: \N_{> 0} \to \N_{> 0}$ be any fixed permutation on $\N_{> 0}$.
Let $\map \det {\mathbf A}$ be the determinant of $\mathbf A$.
Let $\struct {S_n, \circ}$ be the symmetric group of $n$ letters.
Then:
:$\ds \map \det {\mathbf A} = \s... | First it is shown that:
:$\ds \map \det {\mathbf A} = \sum_{\mu \mathop \in S_n} \paren {\map \sgn \mu \map \sgn \lambda \prod_{k \mathop = 1}^n a_{\map \lambda k, \map \mu k} }$
Let $\nu: \N_{> 0} \to \N_{> 0}$ be a permutation on $\N_{> 0}$ such that $\nu \circ \lambda = \mu$.
The product can be rearranged as:
:$\ds ... | Let $\mathbf A = \sqbrk a_n$ be a [[Definition:Square Matrix|square matrix of order $n$]] over a [[Definition:Field (Abstract Algebra)|field]].
Let $\lambda: \N_{> 0} \to \N_{> 0}$ be any fixed [[Definition:Permutation on n Letters|permutation on $\N_{> 0}$]].
Let $\map \det {\mathbf A}$ be the [[Definition:Determina... | First it is shown that:
:$\ds \map \det {\mathbf A} = \sum_{\mu \mathop \in S_n} \paren {\map \sgn \mu \map \sgn \lambda \prod_{k \mathop = 1}^n a_{\map \lambda k, \map \mu k} }$
Let $\nu: \N_{> 0} \to \N_{> 0}$ be a [[Definition:Permutation|permutation]] on $\N_{> 0}$ such that $\nu \circ \lambda = \mu$.
The product... | Permutation of Determinant Indices | https://proofwiki.org/wiki/Permutation_of_Determinant_Indices | https://proofwiki.org/wiki/Permutation_of_Determinant_Indices | [
"Determinants"
] | [
"Definition:Matrix/Square Matrix",
"Definition:Field (Abstract Algebra)",
"Definition:Permutation on n Letters",
"Definition:Determinant/Matrix",
"Definition:Symmetric Group/n Letters",
"Definition:Summation",
"Definition:Permutation on n Letters",
"Definition:Sign of Permutation"
] | [
"Definition:Permutation",
"Parity Function is Homomorphism",
"Definition:Determinant/Matrix",
"Definition:Permutation",
"Category:Determinants"
] |
proofwiki-1407 | Determinant of Transpose | Let $\mathbf A = \sqbrk a_n$ be a square matrix of order $n$.
Let $\map \det {\mathbf A}$ be the determinant of $\mathbf A$.
Let $\mathbf A^\intercal$ be the transpose of $\mathbf A$.
Then:
:$\map \det {\mathbf A} = \map \det {\mathbf A^\intercal}$ | Let $\mathbf A = \begin{bmatrix}
a_{11} & a_{12} & \ldots & a_{1n} \\
a_{21} & a_{22} & \cdots & a_{2n} \\
\vdots & \vdots & \ddots & \vdots \\
a_{n1} & a_{n2} & \cdots & a_{nn} \\
\end{bmatrix}$.
Then $\mathbf A^\intercal = \begin{bmatrix}
a_{11} & a_{21} & \ldots & a_{n1} \\
a_{12} & a_{22} & \cdots & a_{n2} \\
\vdot... | Let $\mathbf A = \sqbrk a_n$ be a [[Definition:Square Matrix|square matrix of order $n$]].
Let $\map \det {\mathbf A}$ be the [[Definition:Determinant of Matrix|determinant]] of $\mathbf A$.
Let $\mathbf A^\intercal$ be the [[Definition:Transpose of Matrix|transpose]] of $\mathbf A$.
Then:
:$\map \det {\mathbf A} =... | Let $\mathbf A = \begin{bmatrix}
a_{11} & a_{12} & \ldots & a_{1n} \\
a_{21} & a_{22} & \cdots & a_{2n} \\
\vdots & \vdots & \ddots & \vdots \\
a_{n1} & a_{n2} & \cdots & a_{nn} \\
\end{bmatrix}$.
Then $\mathbf A^\intercal = \begin{bmatrix}
a_{11} & a_{21} & \ldots & a_{n1} \\
a_{12} & a_{22} & \cdots & a_{n2} \\
\vdo... | Determinant of Transpose | https://proofwiki.org/wiki/Determinant_of_Transpose | https://proofwiki.org/wiki/Determinant_of_Transpose | [
"Determinants",
"Transposes of Matrices"
] | [
"Definition:Matrix/Square Matrix",
"Definition:Determinant/Matrix",
"Definition:Transpose of Matrix"
] | [
"Definition:Determinant/Matrix",
"Permutation of Determinant Indices"
] |
proofwiki-1408 | Determinant with Rows Transposed | If two rows of a matrix with determinant $D$ are transposed, its determinant becomes $-D$. | Let $\mathbf A = \sqbrk a_n$ be a square matrix of order $n$.
Let $1 \le r < s \le n$.
Let $e$ be the elementary row operation that exchanging rows $r$ and $s$.
Let $\mathbf B = \map e {\mathbf A}$.
Let $\mathbf E$ be the elementary row matrix corresponding to $e$.
From Elementary Row Operations as Matrix Multiplicatio... | If two [[Definition:Row of Matrix|rows]] of a [[Definition:Matrix|matrix]] with [[Definition:Determinant of Matrix|determinant]] $D$ are [[Definition:Transposition|transposed]], its [[Definition:Determinant of Matrix|determinant]] becomes $-D$. | Let $\mathbf A = \sqbrk a_n$ be a [[Definition:Square Matrix|square matrix]] of [[Definition:Order of Square Matrix|order $n$]].
Let $1 \le r < s \le n$.
Let $e$ be the [[Definition:Elementary Row Operation|elementary row operation]] that exchanging [[Definition:Row of Matrix|rows]] $r$ and $s$.
Let $\mathbf B = \ma... | Determinant with Rows Transposed/Proof 1 | https://proofwiki.org/wiki/Determinant_with_Rows_Transposed | https://proofwiki.org/wiki/Determinant_with_Rows_Transposed/Proof_1 | [
"Determinant with Rows Transposed",
"Determinants"
] | [
"Definition:Matrix/Row",
"Definition:Matrix",
"Definition:Determinant/Matrix",
"Definition:Transposition",
"Definition:Determinant/Matrix"
] | [
"Definition:Matrix/Square Matrix",
"Definition:Matrix/Square Matrix/Order",
"Definition:Elementary Operation/Row",
"Definition:Matrix/Row",
"Definition:Elementary Matrix/Row Operation",
"Elementary Row Operations as Matrix Multiplications",
"Determinant of Elementary Row Matrix/Exchange Rows",
"Determ... |
proofwiki-1409 | Square Matrix with Duplicate Rows has Zero Determinant | If two rows of a square matrix over a commutative ring $\struct {R, +, \circ}$ are the same, then its determinant is zero. | The proof proceeds by induction over $n$, the order of the square matrix.
=== Basis for the Induction ===
Let $n = 2$, which is the smallest natural number for which a square matrix of order $n$ can have two identical rows.
Let $\mathbf A = \sqbrk a_2$ be a square matrix over $R$ with two identical rows.
Then, by defin... | If two [[Definition:Row of Matrix|rows]] of a [[Definition:Square Matrix|square matrix]] over a [[Definition:Commutative Ring|commutative ring]] $\struct {R, +, \circ}$ are the same, then its [[Definition:Determinant of Matrix|determinant]] is [[Definition:Zero (Number)|zero]]. | The proof proceeds by [[Principle of Mathematical Induction|induction]] over $n$, the [[Definition:Order of Square Matrix|order of the square matrix]].
=== Basis for the Induction ===
Let $n = 2$, which is the smallest [[Definition:Natural Number|natural number]] for which a [[Definition:Square Matrix|square matrix]... | Square Matrix with Duplicate Rows has Zero Determinant/Proof 1 | https://proofwiki.org/wiki/Square_Matrix_with_Duplicate_Rows_has_Zero_Determinant | https://proofwiki.org/wiki/Square_Matrix_with_Duplicate_Rows_has_Zero_Determinant/Proof_1 | [
"Square Matrix with Duplicate Rows has Zero Determinant",
"Determinants",
"Matrix Algebra"
] | [
"Definition:Matrix/Row",
"Definition:Matrix/Square Matrix",
"Definition:Commutative Ring",
"Definition:Determinant/Matrix",
"Definition:Zero (Number)"
] | [
"Principle of Mathematical Induction",
"Definition:Matrix/Square Matrix/Order",
"Definition:Natural Numbers",
"Definition:Matrix/Square Matrix",
"Definition:Matrix/Square Matrix/Order",
"Definition:Matrix/Row",
"Definition:Matrix/Square Matrix",
"Definition:Matrix/Row",
"Definition:Determinant/Matri... |
proofwiki-1410 | Square Matrix with Duplicate Rows has Zero Determinant | If two rows of a square matrix over a commutative ring $\struct {R, +, \circ}$ are the same, then its determinant is zero. | Suppose that $\forall x \in R: x + x = 0 \implies x = 0$.
From Determinant with Rows Transposed, if you exchange two rows of a square matrix, the sign of its determinant changes.
If you exchange two identical rows of a square matrix, then the sign of its determinant changes from $D$, say, to $-D$.
But the matrix stays ... | If two [[Definition:Row of Matrix|rows]] of a [[Definition:Square Matrix|square matrix]] over a [[Definition:Commutative Ring|commutative ring]] $\struct {R, +, \circ}$ are the same, then its [[Definition:Determinant of Matrix|determinant]] is [[Definition:Zero (Number)|zero]]. | Suppose that $\forall x \in R: x + x = 0 \implies x = 0$.
From [[Determinant with Rows Transposed]], if you exchange two [[Definition:Row of Matrix|rows]] of a [[Definition:Square Matrix|square matrix]], the sign of its [[Definition:Determinant of Matrix|determinant]] changes.
If you exchange two identical [[Definiti... | Square Matrix with Duplicate Rows has Zero Determinant/Proof 2 | https://proofwiki.org/wiki/Square_Matrix_with_Duplicate_Rows_has_Zero_Determinant | https://proofwiki.org/wiki/Square_Matrix_with_Duplicate_Rows_has_Zero_Determinant/Proof_2 | [
"Square Matrix with Duplicate Rows has Zero Determinant",
"Determinants",
"Matrix Algebra"
] | [
"Definition:Matrix/Row",
"Definition:Matrix/Square Matrix",
"Definition:Commutative Ring",
"Definition:Determinant/Matrix",
"Definition:Zero (Number)"
] | [
"Determinant with Rows Transposed",
"Definition:Matrix/Row",
"Definition:Matrix/Square Matrix",
"Definition:Determinant/Matrix",
"Definition:Matrix/Row",
"Definition:Matrix/Square Matrix",
"Definition:Determinant/Matrix",
"Definition:Matrix/Square Matrix"
] |
proofwiki-1411 | Determinant with Row Multiplied by Constant | Let $\mathbf A = \sqbrk a_n$ be a square matrix of order $n$.
Let $\map \det {\mathbf A}$ be the determinant of $\mathbf A$.
Let $\mathbf B$ be the matrix resulting from one row of $\mathbf A$ having been multiplied by a constant $c$.
Then:
:$\map \det {\mathbf B} = c \map \det {\mathbf A}$
That is, multiplying one row... | Let $\mathbf A = \sqbrk a_n$ be a square matrix of order $n$.
Let $e$ be the elementary row operation that multiplies rows $i$ by the scalar$c$.
Let $\mathbf B = \map e {\mathbf A}$.
Let $\mathbf E$ be the elementary row matrix corresponding to $e$.
From Elementary Row Operations as Matrix Multiplications:
:$\mathbf B ... | Let $\mathbf A = \sqbrk a_n$ be a [[Definition:Square Matrix|square matrix of order $n$]].
Let $\map \det {\mathbf A}$ be the [[Definition:Determinant of Matrix|determinant]] of $\mathbf A$.
Let $\mathbf B$ be the [[Definition:Square Matrix|matrix]] resulting from one [[Definition:Row of Matrix|row]] of $\mathbf A$ ... | Let $\mathbf A = \sqbrk a_n$ be a [[Definition:Square Matrix|square matrix]] of [[Definition:Order of Square Matrix|order $n$]].
Let $e$ be the [[Definition:Elementary Row Operation|elementary row operation]] that [[Definition:Matrix Scalar Product|multiplies]] [[Definition:Row of Matrix|rows]] $i$ by the [[Definition... | Determinant with Row Multiplied by Constant/Proof 1 | https://proofwiki.org/wiki/Determinant_with_Row_Multiplied_by_Constant | https://proofwiki.org/wiki/Determinant_with_Row_Multiplied_by_Constant/Proof_1 | [
"Determinants",
"Determinant with Row Multiplied by Constant"
] | [
"Definition:Matrix/Square Matrix",
"Definition:Determinant/Matrix",
"Definition:Matrix/Square Matrix",
"Definition:Matrix/Row",
"Definition:Constant",
"Definition:Matrix/Row",
"Definition:Matrix/Square Matrix",
"Definition:Constant",
"Definition:Determinant/Matrix",
"Definition:Constant"
] | [
"Definition:Matrix/Square Matrix",
"Definition:Matrix/Square Matrix/Order",
"Definition:Elementary Operation/Row",
"Definition:Matrix Scalar Product",
"Definition:Matrix/Row",
"Definition:Scalar (Matrix Theory)",
"Definition:Elementary Matrix/Row Operation",
"Elementary Row Operations as Matrix Multip... |
proofwiki-1412 | Determinant with Row Multiplied by Constant | Let $\mathbf A = \sqbrk a_n$ be a square matrix of order $n$.
Let $\map \det {\mathbf A}$ be the determinant of $\mathbf A$.
Let $\mathbf B$ be the matrix resulting from one row of $\mathbf A$ having been multiplied by a constant $c$.
Then:
:$\map \det {\mathbf B} = c \map \det {\mathbf A}$
That is, multiplying one row... | Let:
:<nowiki>$\mathbf A = \begin {bmatrix}
a_{1 1} & a_{1 2} & \cdots & a_{1 n} \\
a_{2 1} & a_{2 2} & \cdots & a_{2 n} \\
\vdots & \vdots & \ddots & \vdots \\
a_{r 1} & a_{r 2} & \cdots & a_{r n} \\
\vdots & \vdots & \ddots & \vdots \\
a_{n 1} & a_{n 2} & \cdots & a_{n n} \\
\end {bmatrix}$</nowiki>
:<nowiki>$\... | Let $\mathbf A = \sqbrk a_n$ be a [[Definition:Square Matrix|square matrix of order $n$]].
Let $\map \det {\mathbf A}$ be the [[Definition:Determinant of Matrix|determinant]] of $\mathbf A$.
Let $\mathbf B$ be the [[Definition:Square Matrix|matrix]] resulting from one [[Definition:Row of Matrix|row]] of $\mathbf A$ ... | Let:
:<nowiki>$\mathbf A = \begin {bmatrix}
a_{1 1} & a_{1 2} & \cdots & a_{1 n} \\
a_{2 1} & a_{2 2} & \cdots & a_{2 n} \\
\vdots & \vdots & \ddots & \vdots \\
a_{r 1} & a_{r 2} & \cdots & a_{r n} \\
\vdots & \vdots & \ddots & \vdots \\
a_{n 1} & a_{n 2} & \cdots & a_{n n} \\
\end {bmatrix}$</nowiki>
:<nowiki>$... | Determinant with Row Multiplied by Constant/Proof 2 | https://proofwiki.org/wiki/Determinant_with_Row_Multiplied_by_Constant | https://proofwiki.org/wiki/Determinant_with_Row_Multiplied_by_Constant/Proof_2 | [
"Determinants",
"Determinant with Row Multiplied by Constant"
] | [
"Definition:Matrix/Square Matrix",
"Definition:Determinant/Matrix",
"Definition:Matrix/Square Matrix",
"Definition:Matrix/Row",
"Definition:Constant",
"Definition:Matrix/Row",
"Definition:Matrix/Square Matrix",
"Definition:Constant",
"Definition:Determinant/Matrix",
"Definition:Constant"
] | [
"Definition:Determinant/Matrix",
"Definition:Summation"
] |
proofwiki-1413 | Determinant as Sum of Determinants | Let $\begin{vmatrix}
a_{11} & \cdots & a_{1s} & \cdots & a_{1n} \\
\vdots & \ddots & \vdots & \ddots & \vdots \\
a_{r1} & \cdots & a_{rs} & \cdots & a_{rn} \\
\vdots & \ddots & \vdots & \ddots & \vdots \\
a_{n1} & \cdots & a_{ns} & \cdots & a_{nn}
\end{vmatrix}$ be a determinant.
Then $\begin{vmatrix}
a_{11} & \cdots &... | Let:
$\quad B = \begin{vmatrix}
a_{11} & \cdots & a_{1s} & \cdots & a_{1n} \\
\vdots & \ddots & \vdots & \ddots & \vdots \\
a_{r1} + a'_{r1} & \cdots & a_{rs} + a'_{rs} & \cdots & a_{rn} + a'_{rn} \\
\vdots & \ddots & \vdots & \ddots & \vdots \\
a_{n1} & \cdots & a_{ns} & \cdots & a_{nn}
\end{vmatrix} = \begin{vmatrix}... | Let $\begin{vmatrix}
a_{11} & \cdots & a_{1s} & \cdots & a_{1n} \\
\vdots & \ddots & \vdots & \ddots & \vdots \\
a_{r1} & \cdots & a_{rs} & \cdots & a_{rn} \\
\vdots & \ddots & \vdots & \ddots & \vdots \\
a_{n1} & \cdots & a_{ns} & \cdots & a_{nn}
\end{vmatrix}$ be a [[Definition:Determinant of Matrix|determinant]].
T... | Let:
$\quad B = \begin{vmatrix}
a_{11} & \cdots & a_{1s} & \cdots & a_{1n} \\
\vdots & \ddots & \vdots & \ddots & \vdots \\
a_{r1} + a'_{r1} & \cdots & a_{rs} + a'_{rs} & \cdots & a_{rn} + a'_{rn} \\
\vdots & \ddots & \vdots & \ddots & \vdots \\
a_{n1} & \cdots & a_{ns} & \cdots & a_{nn}
\end{vmatrix} = \begin{vmatrix... | Determinant as Sum of Determinants | https://proofwiki.org/wiki/Determinant_as_Sum_of_Determinants | https://proofwiki.org/wiki/Determinant_as_Sum_of_Determinants | [
"Determinants"
] | [
"Definition:Determinant/Matrix"
] | [
"Determinant of Transpose",
"Category:Determinants"
] |
proofwiki-1414 | Multiple of Row Added to Row of Determinant | Let <nowiki>$\mathbf A = \begin {bmatrix}
a_{1 1} & a_{1 2} & \cdots & a_{1 n} \\
\vdots & \vdots & \ddots & \vdots \\
a_{r 1} & a_{r 2} & \cdots & a_{r n} \\
\vdots & \vdots & \ddots & \vdots \\
a_{s 1} & a_{s 2} & \cdots & a_{s n} \\
\vdots & \vdots & \ddots & \vdots \\
a_{n 1} & a_{n 2} & \cdots & a_{n n} \... | Let $e$ be the elementary row operation that adds $k$ times row $r$ to row $s$.
Let $\mathbf B = \map e {\mathbf A}$.
Let $\mathbf E$ be the elementary row matrix corresponding to $e$.
From Elementary Row Operations as Matrix Multiplications:
:$\mathbf B = \mathbf E \mathbf A$
From Determinant of Elementary Row Matrix:... | Let <nowiki>$\mathbf A = \begin {bmatrix}
a_{1 1} & a_{1 2} & \cdots & a_{1 n} \\
\vdots & \vdots & \ddots & \vdots \\
a_{r 1} & a_{r 2} & \cdots & a_{r n} \\
\vdots & \vdots & \ddots & \vdots \\
a_{s 1} & a_{s 2} & \cdots & a_{s n} \\
\vdots & \vdots & \ddots & \vdots \\
a_{n 1} & a_{n 2} & \cdots & a_{n n} \... | Let $e$ be the [[Definition:Elementary Row Operation|elementary row operation]] that adds $k$ [[Definition:Matrix Scalar Product|times]] [[Definition:Row of Matrix|row]] $r$ to [[Definition:Row of Matrix|row]] $s$.
Let $\mathbf B = \map e {\mathbf A}$.
Let $\mathbf E$ be the [[Definition:Elementary Row Matrix|element... | Multiple of Row Added to Row of Determinant/Proof 1 | https://proofwiki.org/wiki/Multiple_of_Row_Added_to_Row_of_Determinant | https://proofwiki.org/wiki/Multiple_of_Row_Added_to_Row_of_Determinant/Proof_1 | [
"Multiple of Row Added to Row of Determinant",
"Determinants"
] | [
"Definition:Matrix/Square Matrix",
"Definition:Matrix/Square Matrix/Order",
"Definition:Determinant/Matrix",
"Definition:Determinant/Matrix",
"Definition:Constant",
"Definition:Matrix/Row",
"Definition:Matrix/Row"
] | [
"Definition:Elementary Operation/Row",
"Definition:Matrix Scalar Product",
"Definition:Matrix/Row",
"Definition:Matrix/Row",
"Definition:Elementary Matrix/Row Operation",
"Elementary Row Operations as Matrix Multiplications",
"Determinant of Elementary Row Matrix/Scale Row and Add",
"Determinant of Ma... |
proofwiki-1415 | Multiple of Row Added to Row of Determinant | Let <nowiki>$\mathbf A = \begin {bmatrix}
a_{1 1} & a_{1 2} & \cdots & a_{1 n} \\
\vdots & \vdots & \ddots & \vdots \\
a_{r 1} & a_{r 2} & \cdots & a_{r n} \\
\vdots & \vdots & \ddots & \vdots \\
a_{s 1} & a_{s 2} & \cdots & a_{s n} \\
\vdots & \vdots & \ddots & \vdots \\
a_{n 1} & a_{n 2} & \cdots & a_{n n} \... | By Determinant as Sum of Determinants:
:<nowiki>$\map \det {\mathbf B} = \begin{vmatrix}
a_{11} & a_{12} & \ldots & a_{1n} \\
\vdots & \vdots & \ddots & \vdots \\
a_{r1} + k a_{s1} & a_{r2} + k a_{s2} & \cdots & a_{rn} + k a_{sn} \\
\vdots & \vdots & \ddots & \vdots \\
a_{s1} & a_{s2} & \cdots & a_{sn} \\
\vdots & \vdo... | Let <nowiki>$\mathbf A = \begin {bmatrix}
a_{1 1} & a_{1 2} & \cdots & a_{1 n} \\
\vdots & \vdots & \ddots & \vdots \\
a_{r 1} & a_{r 2} & \cdots & a_{r n} \\
\vdots & \vdots & \ddots & \vdots \\
a_{s 1} & a_{s 2} & \cdots & a_{s n} \\
\vdots & \vdots & \ddots & \vdots \\
a_{n 1} & a_{n 2} & \cdots & a_{n n} \... | By [[Determinant as Sum of Determinants]]:
:<nowiki>$\map \det {\mathbf B} = \begin{vmatrix}
a_{11} & a_{12} & \ldots & a_{1n} \\
\vdots & \vdots & \ddots & \vdots \\
a_{r1} + k a_{s1} & a_{r2} + k a_{s2} & \cdots & a_{rn} + k a_{sn} \\
\vdots & \vdots & \ddots & \vdots \\
a_{s1} & a_{s2} & \cdots & a_{sn} \\
\vdots &... | Multiple of Row Added to Row of Determinant/Proof 2 | https://proofwiki.org/wiki/Multiple_of_Row_Added_to_Row_of_Determinant | https://proofwiki.org/wiki/Multiple_of_Row_Added_to_Row_of_Determinant/Proof_2 | [
"Multiple of Row Added to Row of Determinant",
"Determinants"
] | [
"Definition:Matrix/Square Matrix",
"Definition:Matrix/Square Matrix/Order",
"Definition:Determinant/Matrix",
"Definition:Determinant/Matrix",
"Definition:Constant",
"Definition:Matrix/Row",
"Definition:Matrix/Row"
] | [
"Determinant as Sum of Determinants",
"Determinant with Row Multiplied by Constant",
"Square Matrix with Duplicate Rows has Zero Determinant"
] |
proofwiki-1416 | Determinant of Matrix Product | Let $\mathbf A = \sqbrk a_n$ and $\mathbf B = \sqbrk b_n$ be a square matrices of order $n$.
Let $\map \det {\mathbf A}$ be the determinant of $\mathbf A$.
Let $\mathbf A \mathbf B$ be the (conventional) matrix product of $\mathbf A$ and $\mathbf B$.
Then:
:$\map \det {\mathbf A \mathbf B} = \map \det {\mathbf A} \map ... | This proof assumes that $\mathbf A$ and $\mathbf B$ are $n \times n$-matrices over a commutative ring with unity $\struct {R, +, \circ}$.
Let $\mathbf C = \sqbrk c_n = \mathbf A \mathbf B$.
From Square Matrix is Row Equivalent to Triangular Matrix, it follows that $\mathbf A$ can be converted into a upper triangular ma... | Let $\mathbf A = \sqbrk a_n$ and $\mathbf B = \sqbrk b_n$ be a [[Definition:Square Matrix|square matrices]] of [[Definition:Order of Square Matrix|order $n$]].
Let $\map \det {\mathbf A}$ be the [[Definition:Determinant of Matrix|determinant]] of $\mathbf A$.
Let $\mathbf A \mathbf B$ be the [[Definition:Matrix Produ... | This proof assumes that $\mathbf A$ and $\mathbf B$ are $n \times n$-[[Definition:Matrix|matrices]] over a [[Definition:Commutative and Unitary Ring|commutative ring with unity]] $\struct {R, +, \circ}$.
Let $\mathbf C = \sqbrk c_n = \mathbf A \mathbf B$.
From [[Square Matrix is Row Equivalent to Triangular Matrix]]... | Determinant of Matrix Product/Proof 1 | https://proofwiki.org/wiki/Determinant_of_Matrix_Product | https://proofwiki.org/wiki/Determinant_of_Matrix_Product/Proof_1 | [
"Determinant of Matrix Product",
"Determinants",
"Conventional Matrix Multiplication"
] | [
"Definition:Matrix/Square Matrix",
"Definition:Matrix/Square Matrix/Order",
"Definition:Determinant/Matrix",
"Definition:Matrix Product (Conventional)",
"Definition:Determinant/Matrix",
"Definition:Matrix Product (Conventional)",
"Definition:Multiplication",
"Definition:Determinant/Matrix"
] | [
"Definition:Matrix",
"Definition:Commutative and Unitary Ring",
"Square Matrix is Row Equivalent to Triangular Matrix",
"Definition:Triangular Matrix/Upper Triangular Matrix",
"Definition:Finite Sequence",
"Definition:Elementary Operation/Row",
"Elementary Row Operations Associate with Matrix Multiplica... |
proofwiki-1417 | Determinant of Matrix Product | Let $\mathbf A = \sqbrk a_n$ and $\mathbf B = \sqbrk b_n$ be a square matrices of order $n$.
Let $\map \det {\mathbf A}$ be the determinant of $\mathbf A$.
Let $\mathbf A \mathbf B$ be the (conventional) matrix product of $\mathbf A$ and $\mathbf B$.
Then:
:$\map \det {\mathbf A \mathbf B} = \map \det {\mathbf A} \map ... | Consider two cases:
:$(1): \quad \mathbf A$ is singular.
:$(2): \quad \mathbf A$ is nonsingular.
=== Proof of case $1$ ===
Assume $\mathbf A$ is singular.
Then:
:$\map \det {\mathbf A} = 0$
Also if $\mathbf A$ is singular then so is $\mathbf A \mathbf B$.
Indeed, if $\mathbf A \mathbf B$ has an inverse $\mathbf C$, th... | Let $\mathbf A = \sqbrk a_n$ and $\mathbf B = \sqbrk b_n$ be a [[Definition:Square Matrix|square matrices]] of [[Definition:Order of Square Matrix|order $n$]].
Let $\map \det {\mathbf A}$ be the [[Definition:Determinant of Matrix|determinant]] of $\mathbf A$.
Let $\mathbf A \mathbf B$ be the [[Definition:Matrix Produ... | Consider two cases:
:$(1): \quad \mathbf A$ is [[Definition:Singular Matrix|singular]].
:$(2): \quad \mathbf A$ is [[Definition:Nonsingular Matrix|nonsingular]].
=== Proof of case $1$ ===
Assume $\mathbf A$ is [[Definition:Singular Matrix|singular]].
Then:
:$\map \det {\mathbf A} = 0$
Also if $\mathbf A$ is [[D... | Determinant of Matrix Product/Proof 2 | https://proofwiki.org/wiki/Determinant_of_Matrix_Product | https://proofwiki.org/wiki/Determinant_of_Matrix_Product/Proof_2 | [
"Determinant of Matrix Product",
"Determinants",
"Conventional Matrix Multiplication"
] | [
"Definition:Matrix/Square Matrix",
"Definition:Matrix/Square Matrix/Order",
"Definition:Determinant/Matrix",
"Definition:Matrix Product (Conventional)",
"Definition:Determinant/Matrix",
"Definition:Matrix Product (Conventional)",
"Definition:Multiplication",
"Definition:Determinant/Matrix"
] | [
"Definition:Singular Matrix",
"Definition:Nonsingular Matrix",
"Definition:Singular Matrix",
"Definition:Singular Matrix",
"Left or Right Inverse of Matrix is Inverse",
"Definition:Nonsingular Matrix",
"Definition:Elementary Matrix/Row Operation",
"Definition:Matrix/Square Matrix",
"Definition:Matri... |
proofwiki-1418 | Determinant of Matrix Product | Let $\mathbf A = \sqbrk a_n$ and $\mathbf B = \sqbrk b_n$ be a square matrices of order $n$.
Let $\map \det {\mathbf A}$ be the determinant of $\mathbf A$.
Let $\mathbf A \mathbf B$ be the (conventional) matrix product of $\mathbf A$ and $\mathbf B$.
Then:
:$\map \det {\mathbf A \mathbf B} = \map \det {\mathbf A} \map ... | The Cauchy-Binet Formula gives:
:$\ds \map \det {\mathbf A \mathbf B} = \sum_{1 \mathop \le j_1 \mathop < j_2 \mathop < \cdots \mathop < j_m \le n} \map \det {\mathbf A_{j_1 j_2 \ldots j_m} } \map \det {\mathbf B_{j_1 j_2 \ldots j_m} }$
where:
:$\mathbf A$ is an $m \times n$ matrix
:$\mathbf B$ is an $n \times m$ matri... | Let $\mathbf A = \sqbrk a_n$ and $\mathbf B = \sqbrk b_n$ be a [[Definition:Square Matrix|square matrices]] of [[Definition:Order of Square Matrix|order $n$]].
Let $\map \det {\mathbf A}$ be the [[Definition:Determinant of Matrix|determinant]] of $\mathbf A$.
Let $\mathbf A \mathbf B$ be the [[Definition:Matrix Produ... | The [[Cauchy-Binet Formula]] gives:
:$\ds \map \det {\mathbf A \mathbf B} = \sum_{1 \mathop \le j_1 \mathop < j_2 \mathop < \cdots \mathop < j_m \le n} \map \det {\mathbf A_{j_1 j_2 \ldots j_m} } \map \det {\mathbf B_{j_1 j_2 \ldots j_m} }$
where:
:$\mathbf A$ is an [[Definition:Matrix|$m \times n$ matrix]]
:$\mathbf ... | Determinant of Matrix Product/Proof 3 | https://proofwiki.org/wiki/Determinant_of_Matrix_Product | https://proofwiki.org/wiki/Determinant_of_Matrix_Product/Proof_3 | [
"Determinant of Matrix Product",
"Determinants",
"Conventional Matrix Multiplication"
] | [
"Definition:Matrix/Square Matrix",
"Definition:Matrix/Square Matrix/Order",
"Definition:Determinant/Matrix",
"Definition:Matrix Product (Conventional)",
"Definition:Determinant/Matrix",
"Definition:Matrix Product (Conventional)",
"Definition:Multiplication",
"Definition:Determinant/Matrix"
] | [
"Cauchy-Binet Formula",
"Definition:Matrix",
"Definition:Matrix",
"Definition:Matrix",
"Definition:Matrix/Column",
"Definition:Matrix",
"Definition:Matrix/Row"
] |
proofwiki-1419 | Determinant of Matrix Product | Let $\mathbf A = \sqbrk a_n$ and $\mathbf B = \sqbrk b_n$ be a square matrices of order $n$.
Let $\map \det {\mathbf A}$ be the determinant of $\mathbf A$.
Let $\mathbf A \mathbf B$ be the (conventional) matrix product of $\mathbf A$ and $\mathbf B$.
Then:
:$\map \det {\mathbf A \mathbf B} = \map \det {\mathbf A} \map ... | Remember that $\det$ can be interpreted as an alternating multilinear map with respect to the columns.
This property is sufficient to prove the theorem as follows.
Let $\mathbf A, \mathbf B$ be two $n \times n$ matrices (with coefficients in a commutative field $\mathbb K$ like $\mathbb R$ or $\mathbb C$).
Let us denot... | Let $\mathbf A = \sqbrk a_n$ and $\mathbf B = \sqbrk b_n$ be a [[Definition:Square Matrix|square matrices]] of [[Definition:Order of Square Matrix|order $n$]].
Let $\map \det {\mathbf A}$ be the [[Definition:Determinant of Matrix|determinant]] of $\mathbf A$.
Let $\mathbf A \mathbf B$ be the [[Definition:Matrix Produ... | Remember that $\det$ can be interpreted as an alternating multilinear map with respect to the columns.
This property is sufficient to prove the theorem as follows.
Let $\mathbf A, \mathbf B$ be two $n \times n$ matrices (with coefficients in a commutative field $\mathbb K$ like $\mathbb R$ or $\mathbb C$).
Let us de... | Determinant of Matrix Product/Proof 4 | https://proofwiki.org/wiki/Determinant_of_Matrix_Product | https://proofwiki.org/wiki/Determinant_of_Matrix_Product/Proof_4 | [
"Determinant of Matrix Product",
"Determinants",
"Conventional Matrix Multiplication"
] | [
"Definition:Matrix/Square Matrix",
"Definition:Matrix/Square Matrix/Order",
"Definition:Determinant/Matrix",
"Definition:Matrix Product (Conventional)",
"Definition:Determinant/Matrix",
"Definition:Matrix Product (Conventional)",
"Definition:Multiplication",
"Definition:Determinant/Matrix"
] | [
"Definition:Sign of Permutation"
] |
proofwiki-1420 | Laplace Expansion Theorem for Determinants | Let $\mathbf A = \sqbrk a_n$ be a square matrix of order $n$.
Let $D = \map \det {\mathbf A}$ be the determinant of $\mathbf A$:
:$\ds \map \det {\mathbf A} := \sum_{\lambda} \paren {\map \sgn \lambda \prod_{k \mathop = 1}^n a_{k \map \lambda k} } = \sum_\lambda \map \sgn \lambda a_{1 \map \lambda 1} a_{2 \map \lambda ... | Because of Determinant of Transpose, it is necessary to prove only one of these identities.
Let:
:<nowiki>$D = \begin {vmatrix}
a_{1 1} & \cdots & a_{1 k} & \cdots & a_{1 n} \\
\vdots & \ddots & \vdots & \ddots & \vdots \\
a_{r 1} & \cdots & a_{r k} & \cdots & a_{r n} \\
\vdots & \ddots & \vdots & \ddots & \vdots... | Let $\mathbf A = \sqbrk a_n$ be a [[Definition:Square Matrix|square matrix]] of [[Definition:Order of Square Matrix|order $n$]].
Let $D = \map \det {\mathbf A}$ be the [[Definition:Determinant of Matrix|determinant]] of $\mathbf A$:
:$\ds \map \det {\mathbf A} := \sum_{\lambda} \paren {\map \sgn \lambda \prod_{k \ma... | Because of [[Determinant of Transpose]], it is necessary to prove only one of these identities.
Let:
:<nowiki>$D = \begin {vmatrix}
a_{1 1} & \cdots & a_{1 k} & \cdots & a_{1 n} \\
\vdots & \ddots & \vdots & \ddots & \vdots \\
a_{r 1} & \cdots & a_{r k} & \cdots & a_{r n} \\
\vdots & \ddots & \vdots & \ddots & ... | Laplace Expansion Theorem for Determinants | https://proofwiki.org/wiki/Laplace_Expansion_Theorem_for_Determinants | https://proofwiki.org/wiki/Laplace_Expansion_Theorem_for_Determinants | [
"Laplace Expansion Theorem for Determinants",
"Determinants",
"Named Theorems"
] | [
"Definition:Matrix/Square Matrix",
"Definition:Matrix/Square Matrix/Order",
"Definition:Determinant/Matrix",
"Definition:Permutation on n Letters",
"Definition:Sign of Permutation",
"Definition:Matrix/Element",
"Definition:Cofactor/Element",
"Definition:Determinant/Matrix",
"Definition:Matrix/Elemen... | [
"Determinant of Transpose",
"Determinant with Row Multiplied by Constant",
"Definition:Matrix/Row",
"Determinant as Sum of Determinants",
"Definition:Determinant/Matrix",
"Definition:Matrix/Row",
"Definition:Matrix/Row",
"Definition:Matrix/Row",
"Definition:Matrix/Row",
"Definition:Matrix/Row",
... |
proofwiki-1421 | Determinant with Unit Element in Otherwise Zero Row | Let $D$ be the determinant:
:<nowiki>$D = \begin {vmatrix}
1 & 0 & \cdots & 0 \\
b_{2 1} & b_{2 2} & \cdots & b_{2 n} \\
\vdots & \vdots & \ddots & \vdots \\
b_{n 1} & b_{n 2} & \cdots & b_{n n}
\end {vmatrix}$</nowiki>
Then:
:<nowiki>$D = \begin {vmatrix}
b_{2 2} & \cdots & b_{2 n} \\
\vdots & \d... | We refer to the elements of:
:<nowiki>$\begin {vmatrix}
1 & 0 & \cdots & 0 \\
b_{2 1} & b_{2 2} & \cdots & b_{2 n} \\
\vdots & \vdots & \ddots & \vdots \\
b_{n 1} & b_{n 2} & \cdots & b_{n n}
\end {vmatrix}$</nowiki>
as $\begin {vmatrix} b_{i j} \end {vmatrix}$.
Thus $b_{1 1} = 1, b_{1 2} = 0, \ldo... | Let $D$ be the [[Definition:Determinant of Matrix|determinant]]:
:<nowiki>$D = \begin {vmatrix}
1 & 0 & \cdots & 0 \\
b_{2 1} & b_{2 2} & \cdots & b_{2 n} \\
\vdots & \vdots & \ddots & \vdots \\
b_{n 1} & b_{n 2} & \cdots & b_{n n}
\end {vmatrix}$</nowiki>
Then:
:<nowiki>$D = \begin {vmatrix}
b_... | We refer to the elements of:
:<nowiki>$\begin {vmatrix}
1 & 0 & \cdots & 0 \\
b_{2 1} & b_{2 2} & \cdots & b_{2 n} \\
\vdots & \vdots & \ddots & \vdots \\
b_{n 1} & b_{n 2} & \cdots & b_{n n}
\end {vmatrix}$</nowiki>
as $\begin {vmatrix} b_{i j} \end {vmatrix}$.
Thus $b_{1 1} = 1, b_{1 2} = 0, \... | Determinant with Unit Element in Otherwise Zero Row | https://proofwiki.org/wiki/Determinant_with_Unit_Element_in_Otherwise_Zero_Row | https://proofwiki.org/wiki/Determinant_with_Unit_Element_in_Otherwise_Zero_Row | [
"Determinants"
] | [
"Definition:Determinant/Matrix"
] | [
"Definition:Determinant/Matrix",
"Definition:Permutation on n Letters",
"Definition:Summation",
"Definition:Permutation on n Letters",
"Definition:Fixed Element under Permutation"
] |
proofwiki-1422 | Laplace's Expansion Theorem | Let $D$ be the determinant of order $n$.
Let $r_1, r_2, \ldots, r_k$ be integers such that:
:$1 \le k < n$
:$1 \le r_1 < r_2 < \cdots < r_k \le n$
Let $\map D {r_1, r_2, \ldots, r_k \mid u_1, u_2, \ldots, u_k}$ be an order-$k$ minor of $D$.
Let $\map {\tilde D} {r_1, r_2, \ldots, r_k \mid u_1, u_2, \ldots, u_k}$ be the... | Let us define $r_{k + 1}, r_{k + 2}, \ldots, r_n$ such that:
:$1 \le r_{k + 1} < r_{k + 2} < \cdots < r_n \le n$
:$\rho = \tuple {r_1, r_2, \ldots, r_n}$ is a permutation on $\N^*_n$.
Let $\sigma = \tuple {s_1, s_2, \ldots, s_n}$ be a permutation on $\N^*_n$.
Then by Permutation of Determinant Indices we have:
{{begin-... | Let $D$ be the [[Definition:Determinant of Matrix|determinant of order $n$]].
Let $r_1, r_2, \ldots, r_k$ be [[Definition:Integer|integers]] such that:
:$1 \le k < n$
:$1 \le r_1 < r_2 < \cdots < r_k \le n$
Let $\map D {r_1, r_2, \ldots, r_k \mid u_1, u_2, \ldots, u_k}$ be an [[Definition:Minor of Determinant|order-$... | Let us define $r_{k + 1}, r_{k + 2}, \ldots, r_n$ such that:
:$1 \le r_{k + 1} < r_{k + 2} < \cdots < r_n \le n$
:$\rho = \tuple {r_1, r_2, \ldots, r_n}$ is a [[Definition:Permutation on n Letters|permutation on $\N^*_n$]].
Let $\sigma = \tuple {s_1, s_2, \ldots, s_n}$ be a [[Definition:Permutation on n Letters|permut... | Laplace's Expansion Theorem | https://proofwiki.org/wiki/Laplace's_Expansion_Theorem | https://proofwiki.org/wiki/Laplace's_Expansion_Theorem | [
"Laplace's Expansion Theorem",
"Determinants"
] | [
"Definition:Determinant/Matrix",
"Definition:Integer",
"Definition:Minor of Determinant",
"Definition:Cofactor/Minor",
"Definition:Matrix/Column"
] | [
"Definition:Permutation on n Letters",
"Definition:Permutation on n Letters",
"Permutation of Determinant Indices",
"Definition:Permutation on n Letters",
"Determinant of Transpose"
] |
proofwiki-1423 | Equality of Polynomials | $f$ and $g$ are equal as polynomials {{iff}} $f$ and $g$ are equal as functions.
Thus we can say $f = g$ without ambiguity as to what it means.
{{explain|In the exposition, the term was "equal as forms", but it has now morphed into "equal as polynomials". Needs to be resolved.}} | {{ProofWanted|Proof missing. Also, I am not sure how general this result can be made. My suspicion is that if a comm. ring with $1$, $R$ has no idempotents save $0$ and $1$, then the result continue to hold, but not sure at the moment.}}
Category:Polynomial Theory
gez7nlokbtddy1lrqogfytpdsav6gri | $f$ and $g$ are equal as polynomials {{iff}} $f$ and $g$ are equal as functions.
Thus we can say $f = g$ without ambiguity as to what it means.
{{explain|In the exposition, the term was "equal as forms", but it has now morphed into "equal as polynomials". Needs to be resolved.}} | {{ProofWanted|Proof missing. Also, I am not sure how general this result can be made. My suspicion is that if a comm. ring with $1$, $R$ has no idempotents save $0$ and $1$, then the result continue to hold, but not sure at the moment.}}
[[Category:Polynomial Theory]]
gez7nlokbtddy1lrqogfytpdsav6gri | Equality of Polynomials | https://proofwiki.org/wiki/Equality_of_Polynomials | https://proofwiki.org/wiki/Equality_of_Polynomials | [
"Polynomial Theory"
] | [] | [
"Category:Polynomial Theory"
] |
proofwiki-1424 | Value of Adjugate of Determinant | Let $D$ be the determinant of order $n$.
Let $D^*$ be the adjugate of $D$.
Then $D^* = D^{n - 1}$. | Let <nowiki>$\mathbf A = \begin {bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\
a_{21} & a_{22} & \cdots & a_{2n} \\
\vdots & \vdots & \ddots & \vdots \\
a_{n1} & a_{n2} & \cdots & a_{nn} \end {bmatrix}$</nowiki> and <nowiki>$\mathbf A^* = \begin {bmatrix} A_{11} & A_{12} & \cdots & A_{1n} \\
A_{21} & A_{22} & \cdots &... | Let $D$ be the [[Definition:Determinant of Matrix|determinant]] of order $n$.
Let $D^*$ be the [[Definition:Adjugate Matrix|adjugate]] of $D$.
Then $D^* = D^{n - 1}$. | Let <nowiki>$\mathbf A = \begin {bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\
a_{21} & a_{22} & \cdots & a_{2n} \\
\vdots & \vdots & \ddots & \vdots \\
a_{n1} & a_{n2} & \cdots & a_{nn} \end {bmatrix}$</nowiki> and <nowiki>$\mathbf A^* = \begin {bmatrix} A_{11} & A_{12} & \cdots & A_{1n} \\
A_{21} & A_{22} & \cdots &... | Value of Adjugate of Determinant | https://proofwiki.org/wiki/Value_of_Adjugate_of_Determinant | https://proofwiki.org/wiki/Value_of_Adjugate_of_Determinant | [
"Determinants"
] | [
"Definition:Determinant/Matrix",
"Definition:Adjugate Matrix"
] | [
"Definition:Transpose of Matrix",
"Definition:Matrix Product (Conventional)",
"Determinant of Diagonal Matrix",
"Determinant of Matrix Product",
"Determinant of Transpose",
"Laplace Expansion Theorem for Determinants",
"Category:Determinants"
] |
proofwiki-1425 | Determinant of Diagonal Matrix | Let $\mathbf A = \begin{bmatrix}
a_{11} & 0 & \cdots & 0 \\
0 & a_{22} & \cdots & 0 \\
\vdots & \vdots & \ddots & \vdots \\
0 & 0 & \cdots & a_{nn} \\
\end{bmatrix}$ be a diagonal matrix.
Then the determinant of $\mathbf A$ is the product of the elements of $\mathbf A$.
That is:
:$\ds \map \det {\mathbf A} = \prod_{i \... | As a diagonal matrix is also a triangular matrix (both upper and lower), the result follows directly from Determinant of Triangular Matrix.
{{qed}}
Category:Determinants
Category:Diagonal Matrices
46j7gt7zea6di1bocalbw6n08ipn7hx | Let $\mathbf A = \begin{bmatrix}
a_{11} & 0 & \cdots & 0 \\
0 & a_{22} & \cdots & 0 \\
\vdots & \vdots & \ddots & \vdots \\
0 & 0 & \cdots & a_{nn} \\
\end{bmatrix}$ be a [[Definition:Diagonal Matrix|diagonal matrix]].
Then the [[Definition:Determinant of Matrix|determinant]] of $\mathbf A$ is the product of the eleme... | As a [[Definition:Diagonal Matrix|diagonal matrix]] is also a [[Definition:Triangular Matrix|triangular matrix]] (both upper and lower), the result follows directly from [[Determinant of Triangular Matrix]].
{{qed}}
[[Category:Determinants]]
[[Category:Diagonal Matrices]]
46j7gt7zea6di1bocalbw6n08ipn7hx | Determinant of Diagonal Matrix | https://proofwiki.org/wiki/Determinant_of_Diagonal_Matrix | https://proofwiki.org/wiki/Determinant_of_Diagonal_Matrix | [
"Determinants",
"Diagonal Matrices"
] | [
"Definition:Diagonal Matrix",
"Definition:Determinant/Matrix"
] | [
"Definition:Diagonal Matrix",
"Definition:Triangular Matrix",
"Determinant of Triangular Matrix",
"Category:Determinants",
"Category:Diagonal Matrices"
] |
proofwiki-1426 | Row Equivalence is Equivalence Relation | Row equivalence is an equivalence relation. | In the following, $\mathbf A$, $\mathbf B$ and $\mathbf C$ denote arbitrary matrices in a given matrix space $\map \MM {m, n}$ for $m, n \in \Z_{>0}$.
We check in turn each of the conditions for equivalence: | [[Definition:Row Equivalence|Row equivalence]] is an [[Definition:Equivalence Relation|equivalence relation]]. | In the following, $\mathbf A$, $\mathbf B$ and $\mathbf C$ denote arbitrary [[Definition:Matrix|matrices]] in a given [[Definition:Matrix Space|matrix space]] $\map \MM {m, n}$ for $m, n \in \Z_{>0}$.
We check in turn each of the conditions for [[Definition:Equivalence Relation|equivalence]]: | Row Equivalence is Equivalence Relation | https://proofwiki.org/wiki/Row_Equivalence_is_Equivalence_Relation | https://proofwiki.org/wiki/Row_Equivalence_is_Equivalence_Relation | [
"Examples of Equivalence Relations",
"Row Operations",
"Row Equivalence"
] | [
"Definition:Row Equivalence",
"Definition:Equivalence Relation"
] | [
"Definition:Matrix",
"Definition:Matrix Space",
"Definition:Equivalence Relation",
"Definition:Equivalence Relation"
] |
proofwiki-1427 | Matrix is Row Equivalent to Reduced Echelon Matrix | Let $\mathbf A = \sqbrk a_{m n}$ be a matrix of order $m \times n$ over a field $F$.
Then $A$ is row equivalent to a reduced echelon matrix of order $m \times n$. | {{Recall|Gaussian Elimination}}
{{:Definition:Gaussian Elimination/Procedure}}
Hence the result.
{{qed}} | Let $\mathbf A = \sqbrk a_{m n}$ be a [[Definition:Matrix|matrix]] of [[Definition:Order of Matrix|order]] $m \times n$ over a [[Definition:Field (Abstract Algebra)|field]] $F$.
Then $A$ is [[Definition:Row Equivalence|row equivalent]] to a [[Definition:Reduced Echelon Matrix|reduced echelon matrix]] of [[Definition:... | {{Recall|Gaussian Elimination}}
{{:Definition:Gaussian Elimination/Procedure}}
Hence the result.
{{qed}} | Matrix is Row Equivalent to Reduced Echelon Matrix/Proof | https://proofwiki.org/wiki/Matrix_is_Row_Equivalent_to_Reduced_Echelon_Matrix | https://proofwiki.org/wiki/Matrix_is_Row_Equivalent_to_Reduced_Echelon_Matrix/Proof | [
"Echelon Matrices"
] | [
"Definition:Matrix",
"Definition:Matrix/Order",
"Definition:Field (Abstract Algebra)",
"Definition:Row Equivalence",
"Definition:Echelon Matrix/Reduced Echelon Form",
"Definition:Matrix/Order"
] | [] |
proofwiki-1428 | Square Matrix is Row Equivalent to Triangular Matrix | Let $\mathbf A = \sqbrk a_n$ be a square matrix of order $n$ over a commutative ring $R$.
Then $\mathbf A$ can be converted to an upper or lower triangular matrix by elementary row operations. | Let $\mathbf A$ be a square matrix of order $n$.
We proceed by induction on $n$, the number of rows of $\mathbf A$.
=== Basis for the Induction ===
For $n = 1$, we have a matrix of just one element, which is trivially diagonal, hence both upper and lower triangular.
This is the basis for the induction.
=== Induction Hy... | Let $\mathbf A = \sqbrk a_n$ be a [[Definition:Square Matrix|square matrix of order $n$]] over a [[Definition:Commutative Ring|commutative ring]] $R$.
Then $\mathbf A$ can be converted to an [[Definition:Upper Triangular Matrix|upper]] or [[Definition:Lower Triangular Matrix|lower triangular matrix]] by [[Definition:E... | Let $\mathbf A$ be a [[Definition:Square Matrix|square matrix of order $n$]].
We proceed by induction on $n$, the number of [[Definition:Row of Matrix|rows]] of $\mathbf A$.
=== Basis for the Induction ===
For $n = 1$, we have a matrix of just one [[Definition:Element of Matrix|element]], which is trivially [[Defin... | Square Matrix is Row Equivalent to Triangular Matrix/Proof 1 | https://proofwiki.org/wiki/Square_Matrix_is_Row_Equivalent_to_Triangular_Matrix | https://proofwiki.org/wiki/Square_Matrix_is_Row_Equivalent_to_Triangular_Matrix/Proof_1 | [
"Square Matrices",
"Triangular Matrices",
"Square Matrix is Row Equivalent to Triangular Matrix"
] | [
"Definition:Matrix/Square Matrix",
"Definition:Commutative Ring",
"Definition:Triangular Matrix/Upper Triangular Matrix",
"Definition:Triangular Matrix/Lower Triangular Matrix",
"Definition:Elementary Operation/Row"
] | [
"Definition:Matrix/Square Matrix",
"Definition:Matrix/Row",
"Definition:Matrix/Element",
"Definition:Diagonal Matrix",
"Definition:Basis for the Induction",
"Definition:Elementary Operation/Row",
"Definition:Field (Abstract Algebra)",
"Definition:Induction Hypothesis",
"Definition:Matrix/Square Matr... |
proofwiki-1429 | Square Matrix is Row Equivalent to Triangular Matrix | Let $\mathbf A = \sqbrk a_n$ be a square matrix of order $n$ over a commutative ring $R$.
Then $\mathbf A$ can be converted to an upper or lower triangular matrix by elementary row operations. | This proof assumes that $R$ is a field, which makes the triangulation process slightly quicker.
By this assumptions, all elements of $\mathbf A$ have multiplicative inverses.
Let $\mathbf A$ be a square matrix of order $n$.
We proceed by induction on $n$, the number of rows of $\mathbf A$.
=== Basis for the Induction =... | Let $\mathbf A = \sqbrk a_n$ be a [[Definition:Square Matrix|square matrix of order $n$]] over a [[Definition:Commutative Ring|commutative ring]] $R$.
Then $\mathbf A$ can be converted to an [[Definition:Upper Triangular Matrix|upper]] or [[Definition:Lower Triangular Matrix|lower triangular matrix]] by [[Definition:E... | This proof assumes that $R$ is a [[Definition:Field (Abstract Algebra)|field]], which makes the triangulation process slightly quicker.
By this assumptions, all [[Definition:Element of Matrix|elements]] of $\mathbf A$ have [[Definition:Inverse Element|multiplicative inverses]].
Let $\mathbf A$ be a [[Definition:Squa... | Square Matrix is Row Equivalent to Triangular Matrix/Proof 2 | https://proofwiki.org/wiki/Square_Matrix_is_Row_Equivalent_to_Triangular_Matrix | https://proofwiki.org/wiki/Square_Matrix_is_Row_Equivalent_to_Triangular_Matrix/Proof_2 | [
"Square Matrices",
"Triangular Matrices",
"Square Matrix is Row Equivalent to Triangular Matrix"
] | [
"Definition:Matrix/Square Matrix",
"Definition:Commutative Ring",
"Definition:Triangular Matrix/Upper Triangular Matrix",
"Definition:Triangular Matrix/Lower Triangular Matrix",
"Definition:Elementary Operation/Row"
] | [
"Definition:Field (Abstract Algebra)",
"Definition:Matrix/Element",
"Definition:Inverse (Abstract Algebra)/Inverse",
"Definition:Matrix/Square Matrix",
"Definition:Matrix/Row",
"Definition:Matrix/Element",
"Definition:Diagonal Matrix",
"Definition:Basis for the Induction",
"Definition:Elementary Ope... |
proofwiki-1430 | Product of Triangular Matrices | Let $\mathbf A = \sqbrk a_n, \mathbf B = \sqbrk b_n$ be upper triangular matrices of order $n$.
Let $\mathbf C = \mathbf A \mathbf B$.
Then
:$(1): \quad$ the diagonal elements of $\mathbf C$ are given by:
::::$\forall j \in \closedint 1 n: c_{j j} = a_{j j} b_{j j}$
:::That is, the diagonal elements of $\mathbf C$ are ... | From the definition of matrix product, we have:
:$\ds \forall i, j \in \closedint 1 n: c_{i j} = \sum_{k \mathop = 1}^n a_{i k} b_{k j}$
Now when $i = j$ (as on the main diagonal):
:$\ds c_{j j} = \sum_{k \mathop = 1}^n a_{j k} b_{k j}$
Now both $\mathbf A$ and $\mathbf B$ are upper triangular.
Thus:
:if $k > j$, then ... | Let $\mathbf A = \sqbrk a_n, \mathbf B = \sqbrk b_n$ be [[Definition:Upper Triangular Matrix|upper triangular matrices]] of [[Definition:Order of Square Matrix|order]] $n$.
Let $\mathbf C = \mathbf A \mathbf B$.
Then
:$(1): \quad$ the [[Definition:Diagonal Element|diagonal elements]] of $\mathbf C$ are given by:
:::... | From the definition of [[Definition:Matrix Product (Conventional)|matrix product]], we have:
:$\ds \forall i, j \in \closedint 1 n: c_{i j} = \sum_{k \mathop = 1}^n a_{i k} b_{k j}$
Now when $i = j$ (as on the [[Definition:Diagonal Element|main diagonal]]):
:$\ds c_{j j} = \sum_{k \mathop = 1}^n a_{j k} b_{k j}$
Now... | Product of Triangular Matrices | https://proofwiki.org/wiki/Product_of_Triangular_Matrices | https://proofwiki.org/wiki/Product_of_Triangular_Matrices | [
"Triangular Matrices",
"Conventional Matrix Multiplication"
] | [
"Definition:Triangular Matrix/Upper Triangular Matrix",
"Definition:Matrix/Square Matrix/Order",
"Definition:Main Diagonal/Diagonal Elements",
"Definition:Main Diagonal/Diagonal Elements",
"Definition:Triangular Matrix/Upper Triangular Matrix",
"Definition:Triangular Matrix/Lower Triangular Matrix"
] | [
"Definition:Matrix Product (Conventional)",
"Definition:Main Diagonal/Diagonal Elements",
"Definition:Triangular Matrix/Upper Triangular Matrix",
"Definition:Zero Element",
"Definition:Triangular Matrix/Upper Triangular Matrix",
"Definition:Triangular Matrix/Lower Triangular Matrix",
"Category:Triangula... |
proofwiki-1431 | Effect of Elementary Row Operations on Determinant | Let $\mathbf A = \sqbrk a_n$ be a square matrix of order $n$.
Let $\map \det {\mathbf A}$ denote the determinant of $\mathbf A$.
Let $i, j \in \closedint 1 n : i \neq j$ be indices of the rows of $\mathbf A$.
Take the elementary row operations:
{{begin-axiom}}
{{axiom | n = \text {ERO} 1
| t = For some $\lambda... | From Elementary Row Operations as Matrix Multiplications, an elementary row operation on $\mathbf A$ is equivalent to matrix multiplication by the elementary row matrices corresponding to the elementary row operations.
From Determinant of Elementary Row Matrix, the determinants of those elementary row matrices are as f... | Let $\mathbf A = \sqbrk a_n$ be a [[Definition:Square Matrix|square matrix of order $n$]].
Let $\map \det {\mathbf A}$ denote the [[Definition:Determinant of Matrix|determinant]] of $\mathbf A$.
Let $i, j \in \closedint 1 n : i \neq j$ be [[Definition:Index of Term of Sequence|indices]] of the [[Definition:Row of Mat... | From [[Elementary Row Operations as Matrix Multiplications]], an [[Definition:Elementary Row Operation|elementary row operation]] on $\mathbf A$ is equivalent to [[Definition:Matrix Product (Conventional)|matrix multiplication]] by the [[Definition:Elementary Row Matrix|elementary row matrices]] corresponding to the [[... | Effect of Elementary Row Operations on Determinant | https://proofwiki.org/wiki/Effect_of_Elementary_Row_Operations_on_Determinant | https://proofwiki.org/wiki/Effect_of_Elementary_Row_Operations_on_Determinant | [
"Determinants",
"Elementary Row Operations"
] | [
"Definition:Matrix/Square Matrix",
"Definition:Determinant/Matrix",
"Definition:Term of Sequence/Index",
"Definition:Matrix/Row",
"Definition:Elementary Operation/Row",
"Definition:Matrix Scalar Product",
"Definition:Matrix/Row",
"Definition:Matrix Scalar Product",
"Definition:Matrix/Row",
"Defini... | [
"Elementary Row Operations as Matrix Multiplications",
"Definition:Elementary Operation/Row",
"Definition:Matrix Product (Conventional)",
"Definition:Elementary Matrix/Row Operation",
"Definition:Elementary Operation/Row",
"Determinant of Elementary Row Matrix",
"Definition:Determinant/Matrix",
"Defin... |
proofwiki-1432 | Modulus in Terms of Conjugate | Let $z = a + i b$ be a complex number.
Let $\cmod z$ be the modulus of $z$.
Let $\overline z$ be the conjugate of $z$.
Then:
:$\cmod z = \sqrt {z \overline z}$ | Let $z = a + i b$.
Then:
{{begin-eqn}}
{{eqn | l = z \overline z
| r = a^2 + b^2
| c = Product of Complex Number with Conjugate
}}
{{eqn | r = \cmod z^2
| c = {{Defof|Complex Modulus}}
}}
{{eqn | ll= \leadsto
| l = \cmod z
| r = \sqrt {z \overline z}
| c = taking square root of both ... | Let $z = a + i b$ be a [[Definition:Complex Number|complex number]].
Let $\cmod z$ be the [[Definition:Complex Modulus|modulus]] of $z$.
Let $\overline z$ be the [[Definition:Complex Conjugate|conjugate]] of $z$.
Then:
:$\cmod z = \sqrt {z \overline z}$ | Let $z = a + i b$.
Then:
{{begin-eqn}}
{{eqn | l = z \overline z
| r = a^2 + b^2
| c = [[Product of Complex Number with Conjugate]]
}}
{{eqn | r = \cmod z^2
| c = {{Defof|Complex Modulus}}
}}
{{eqn | ll= \leadsto
| l = \cmod z
| r = \sqrt {z \overline z}
| c = taking [[Definition:S... | Modulus in Terms of Conjugate | https://proofwiki.org/wiki/Modulus_in_Terms_of_Conjugate | https://proofwiki.org/wiki/Modulus_in_Terms_of_Conjugate | [
"Complex Conjugates",
"Complex Modulus"
] | [
"Definition:Complex Number",
"Definition:Complex Modulus",
"Definition:Complex Conjugate"
] | [
"Product of Complex Number with Conjugate",
"Definition:Square Root"
] |
proofwiki-1433 | Existence and Uniqueness of Positive Root of Positive Real Number | Let $x \in \R$ be a real number such that $x \ge 0$.
Let $n \in \Z$ be an integer such that $n \ne 0$.
Then there always exists a unique $y \in \R: \paren {y \ge 0} \land \paren {y^n = x}$.
Hence the justification for the terminology '''the positive $n$th root of $x$''' and the notation $x^{1/n}$.
=== Positive Exponent... | The result follows from Existence of Positive Root of Positive Real Number and Uniqueness of Positive Root of Positive Real Number.
{{qed}} | Let $x \in \R$ be a [[Definition:Real Number|real number]] such that $x \ge 0$.
Let $n \in \Z$ be an [[Definition:Integer|integer]] such that $n \ne 0$.
Then there always exists a [[Definition:Unique|unique]] $y \in \R: \paren {y \ge 0} \land \paren {y^n = x}$.
Hence the justification for the terminology '''the pos... | The result follows from [[Existence of Positive Root of Positive Real Number]] and [[Uniqueness of Positive Root of Positive Real Number]].
{{qed}} | Existence and Uniqueness of Positive Root of Positive Real Number | https://proofwiki.org/wiki/Existence_and_Uniqueness_of_Positive_Root_of_Positive_Real_Number | https://proofwiki.org/wiki/Existence_and_Uniqueness_of_Positive_Root_of_Positive_Real_Number | [
"Existence and Uniqueness of Positive Root of Positive Real Number",
"Roots of Numbers",
"Real Numbers"
] | [
"Definition:Real Number",
"Definition:Integer",
"Definition:Unique",
"Definition:Root of Number",
"Existence and Uniqueness of Positive Root of Positive Real Number/Positive Exponent",
"Existence and Uniqueness of Positive Root of Positive Real Number/Negative Exponent"
] | [
"Existence of Positive Root of Positive Real Number",
"Uniqueness of Positive Root of Positive Real Number"
] |
proofwiki-1434 | Even Power is Non-Negative | Let $x \in \R$ be a real number.
Let $n \in \Z$ be an even integer.
Then $x^n \ge 0$.
That is, all even powers are positive. | Let $n \in \Z$ be an even integer.
Then $n = 2 k$ for some $k \in \Z$.
Thus:
:$\forall x \in \R: x^n = x^{2 k} = \paren {x^k}^2$
But from Square of Real Number is Non-Negative:
:$\forall x \in \R: \paren {x^k}^2 \ge 0$
and so there is no real number whose square is negative.
The result follows from Solution to Quadrati... | Let $x \in \R$ be a [[Definition:Real Number|real number]].
Let $n \in \Z$ be an [[Definition:Even Integer|even integer]].
Then $x^n \ge 0$.
That is, all [[Definition:Even Power|even powers]] are [[Definition:Positive Real Number|positive]]. | Let $n \in \Z$ be an [[Definition:Even Integer|even integer]].
Then $n = 2 k$ for some $k \in \Z$.
Thus:
:$\forall x \in \R: x^n = x^{2 k} = \paren {x^k}^2$
But from [[Square of Real Number is Non-Negative]]:
:$\forall x \in \R: \paren {x^k}^2 \ge 0$
and so there is no [[Definition:Real Number|real number]] whose [[... | Even Power is Non-Negative | https://proofwiki.org/wiki/Even_Power_is_Non-Negative | https://proofwiki.org/wiki/Even_Power_is_Non-Negative | [
"Powers"
] | [
"Definition:Real Number",
"Definition:Even Integer",
"Definition:Power (Algebra)/Even Power",
"Definition:Positive/Real Number"
] | [
"Definition:Even Integer",
"Square of Real Number is Non-Negative",
"Definition:Real Number",
"Definition:Square/Function",
"Definition:Strictly Negative/Real Number",
"Solution to Quadratic Equation"
] |
proofwiki-1435 | Sign of Odd Power | Let $x \in \R$ be a real number.
Let $n \in \Z$ be an odd integer.
Then:
:$x^n = 0 \iff x = 0$
:$x^n > 0 \iff x > 0$
:$x^n < 0 \iff x < 0$
That is, the sign of an odd power matches the number it is a power of. | If $n$ is an odd integer, then $n = 2 k + 1$ for some $k \in \N$.
Thus $x^n = x \cdot x^{2 k}$.
But $x^{2 k} \ge 0$ from Even Power is Non-Negative.
The result follows.
{{qed}} | Let $x \in \R$ be a [[Definition:Real Number|real number]].
Let $n \in \Z$ be an [[Definition:Odd Integer|odd integer]].
Then:
:$x^n = 0 \iff x = 0$
:$x^n > 0 \iff x > 0$
:$x^n < 0 \iff x < 0$
That is, the sign of an [[Definition:Odd Power|odd power]] matches the number it is a [[Definition:Power (Algebra)|power]]... | If $n$ is an [[Definition:Odd Integer|odd integer]], then $n = 2 k + 1$ for some $k \in \N$.
Thus $x^n = x \cdot x^{2 k}$.
But $x^{2 k} \ge 0$ from [[Even Power is Non-Negative]].
The result follows.
{{qed}} | Sign of Odd Power | https://proofwiki.org/wiki/Sign_of_Odd_Power | https://proofwiki.org/wiki/Sign_of_Odd_Power | [
"Real Analysis"
] | [
"Definition:Real Number",
"Definition:Odd Integer",
"Definition:Power (Algebra)/Odd Power",
"Definition:Power (Algebra)"
] | [
"Definition:Odd Integer",
"Even Power is Non-Negative"
] |
proofwiki-1436 | Product of Absolute Values on Ordered Integral Domain | Let $\struct {D, +, \times, \le}$ be an ordered integral domain whose zero is denoted by $0_D$.
For all $a \in D$, let $\size a$ denote the absolute value of $a$.
Then:
:$\size a \times \size b = \size {a \times b}$ | Let $P$ be the (strict) positivity property on $D$.
Let $<$ be the (strict) total ordering defined on $D$ as:
:$a < b \iff a \le b \land a \ne b$
Let $N$ be the strict negativity property on $D$.
We consider all possibilities in turn.
$(1): \quad a = 0_D$ or $b = 0_D$
In this case, both the {{LHS}} $\size a \times \siz... | Let $\struct {D, +, \times, \le}$ be an [[Definition:Ordered Integral Domain|ordered integral domain]] whose [[Definition:Ring Zero|zero]] is denoted by $0_D$.
For all $a \in D$, let $\size a$ denote the [[Definition:Absolute Value on Ordered Integral Domain|absolute value]] of $a$.
Then:
:$\size a \times \size b = ... | Let $P$ be the [[Definition:Strict Positivity Property|(strict) positivity property]] on $D$.
Let $<$ be the [[Definition:Strict Total Ordering|(strict) total ordering]] defined on $D$ as:
:$a < b \iff a \le b \land a \ne b$
Let $N$ be the [[Definition:Strict Negativity Property|strict negativity property]] on $D$.
... | Product of Absolute Values on Ordered Integral Domain | https://proofwiki.org/wiki/Product_of_Absolute_Values_on_Ordered_Integral_Domain | https://proofwiki.org/wiki/Product_of_Absolute_Values_on_Ordered_Integral_Domain | [
"Absolute Value Function",
"Ordered Integral Domains"
] | [
"Definition:Ordered Integral Domain",
"Definition:Ring Zero",
"Definition:Absolute Value/Ordered Integral Domain"
] | [
"Definition:Strict Positivity Property",
"Definition:Strict Total Ordering",
"Definition:Strict Negativity Property",
"Definition:Ring Zero",
"Definition:Strict Positivity Property",
"Product with Ring Negative",
"Properties of Strict Negativity",
"Product of Ring Negatives",
"Properties of Strict N... |
proofwiki-1437 | Negative of Absolute Value | Let $x \in \R$ be a real number.
Let $\size x$ denote the absolute value of $x$.
Then:
:$-\size x \le x \le \size x$ | Either $x \ge 0$ or $x < 0$.
:If $x \ge 0$, then:
::$-\size x \le 0 \le x = \size x$
:If $x < 0$, then:
::$-\size x = x < 0 < \size x$
{{qed}} | Let $x \in \R$ be a [[Definition:Real Number|real number]].
Let $\size x$ denote the [[Definition:Absolute Value|absolute value]] of $x$.
Then:
:$-\size x \le x \le \size x$ | Either $x \ge 0$ or $x < 0$.
:If $x \ge 0$, then:
::$-\size x \le 0 \le x = \size x$
:If $x < 0$, then:
::$-\size x = x < 0 < \size x$
{{qed}} | Negative of Absolute Value | https://proofwiki.org/wiki/Negative_of_Absolute_Value | https://proofwiki.org/wiki/Negative_of_Absolute_Value | [
"Negative of Absolute Value",
"Absolute Value Function",
"Inequalities"
] | [
"Definition:Real Number",
"Definition:Absolute Value"
] | [] |
proofwiki-1438 | Order of Squares in Ordered Ring | Let $\struct {R, +, \circ, \le}$ be an ordered ring whose zero is $0_R$ and whose unity is $1_R$.
Let $x, y \in \struct {R, +, \circ, \le}$ such that $0_R \le x, y$.
Then:
:$x \le y \implies x \circ x \le y \circ y$
When $R$ is one of the standard sets of numbers, that is $\Z, \Q, \R$, then this translates into:
:If $x... | Assume $x \le y$.
As $\le$ is compatible with the ring structure of $\struct {R, +, \circ, \le}$, we have:
:$x \ge 0 \implies x \circ x \le x \circ y$
:$y \ge 0 \implies x \circ y \le y \circ y$
and thus as $\le$ is transitive, it follows that $x \circ x \le y \circ y$.
{{qed}} | Let $\struct {R, +, \circ, \le}$ be an [[Definition:Ordered Ring|ordered ring]] whose [[Definition:Ring Zero|zero]] is $0_R$ and whose [[Definition:Unity of Ring|unity]] is $1_R$.
Let $x, y \in \struct {R, +, \circ, \le}$ such that $0_R \le x, y$.
Then:
:$x \le y \implies x \circ x \le y \circ y$
When $R$ is one o... | Assume $x \le y$.
As $\le$ is [[Definition:Ordering Compatible with Ring Structure|compatible]] with the ring structure of $\struct {R, +, \circ, \le}$, we have:
:$x \ge 0 \implies x \circ x \le x \circ y$
:$y \ge 0 \implies x \circ y \le y \circ y$
and thus as $\le$ is [[Definition:Ordering|transitive]], it follows... | Order of Squares in Ordered Ring | https://proofwiki.org/wiki/Order_of_Squares_in_Ordered_Ring | https://proofwiki.org/wiki/Order_of_Squares_in_Ordered_Ring | [
"Ordered Rings"
] | [
"Definition:Ordered Ring",
"Definition:Ring Zero",
"Definition:Unity (Abstract Algebra)/Ring",
"Definition:Number",
"Definition:Positive"
] | [
"Definition:Ordering Compatible with Ring Structure",
"Definition:Ordering"
] |
proofwiki-1439 | Least Upper Bound Property | Let $S \subset \R$ be a non-empty subset of the set of real numbers such that $S$ is bounded above.
Then $S$ admits a supremum in $\R$.
This is known as the '''least upper bound property''' of the real numbers. | Suppose that $S \subseteq \R_{\ge 0}$ has the positive real number $U$ as an upper bound.
Then $\R_{\ge 0}$ can be represented as a straight line $L$ whose sole endpoint is the point $O$.
Let $l_0 \in \R_{\ge 0}$ be the standard unit of length.
There exists a unique point $X \in L$ such that $U \cdot l_0 = OX$.
Furthe... | Let $S \subset \R$ be a [[Definition:Non-Empty Set|non-empty]] [[Definition:Subset|subset]] of the [[Definition:Real Number|set of real numbers]] such that $S$ is [[Definition:Bounded Above Subset of Real Numbers|bounded above]].
Then $S$ [[Definition:Supremum of Subset of Real Numbers|admits a supremum]] in $\R$.
Th... | Suppose that $S \subseteq \R_{\ge 0}$ has the [[Definition:Positive Real Number|positive real number]] $U$ as an [[Definition:Upper Bound of Subset of Real Numbers|upper bound]].
Then $\R_{\ge 0}$ can be represented as a [[Definition:Real Number Line|straight line]] $L$ whose sole [[Definition:Endpoint of Line|endpoin... | Least Upper Bound Property/Proof 1 | https://proofwiki.org/wiki/Least_Upper_Bound_Property | https://proofwiki.org/wiki/Least_Upper_Bound_Property/Proof_1 | [
"Least Upper Bound Property",
"Continuum Property",
"Named Theorems"
] | [
"Definition:Non-Empty Set",
"Definition:Subset",
"Definition:Real Number",
"Definition:Bounded Above Set/Real Numbers",
"Definition:Supremum of Set/Real Numbers",
"Least Upper Bound Property",
"Definition:Real Number"
] | [
"Definition:Positive/Real Number",
"Definition:Upper Bound of Set/Real Numbers",
"Definition:Real Number/Real Number Line",
"Definition:Line/Endpoint",
"Definition:Point",
"Definition:Unit of Measurement",
"Definition:Linear Measure/Length",
"Definition:Unique",
"Definition:Point",
"Definition:Mul... |
proofwiki-1440 | Least Upper Bound Property | Let $S \subset \R$ be a non-empty subset of the set of real numbers such that $S$ is bounded above.
Then $S$ admits a supremum in $\R$.
This is known as the '''least upper bound property''' of the real numbers. | Let $S$ be bounded above.
Let $L$ be the set of real numbers defined as:
:$\alpha \in L \iff \exists x \in S: \alpha < x$
Let $R := \relcomp \R L$, where $\complement_\R$ denotes complement in $\R$.
By construction of $L$, every element of $L$ is less than some element of $S$.
Hence no element of $L$ is an upper bound ... | Let $S \subset \R$ be a [[Definition:Non-Empty Set|non-empty]] [[Definition:Subset|subset]] of the [[Definition:Real Number|set of real numbers]] such that $S$ is [[Definition:Bounded Above Subset of Real Numbers|bounded above]].
Then $S$ [[Definition:Supremum of Subset of Real Numbers|admits a supremum]] in $\R$.
Th... | Let $S$ be [[Definition:Bounded Above Subset of Real Numbers|bounded above]].
Let $L$ be the [[Definition:Set|set]] of [[Definition:Real Number|real numbers]] defined as:
:$\alpha \in L \iff \exists x \in S: \alpha < x$
Let $R := \relcomp \R L$, where $\complement_\R$ denotes [[Definition:Relative Complement|compleme... | Least Upper Bound Property/Proof 2 | https://proofwiki.org/wiki/Least_Upper_Bound_Property | https://proofwiki.org/wiki/Least_Upper_Bound_Property/Proof_2 | [
"Least Upper Bound Property",
"Continuum Property",
"Named Theorems"
] | [
"Definition:Non-Empty Set",
"Definition:Subset",
"Definition:Real Number",
"Definition:Bounded Above Set/Real Numbers",
"Definition:Supremum of Set/Real Numbers",
"Least Upper Bound Property",
"Definition:Real Number"
] | [
"Definition:Bounded Above Set/Real Numbers",
"Definition:Set",
"Definition:Real Number",
"Definition:Relative Complement",
"Definition:Element",
"Definition:Element",
"Definition:Element",
"Definition:Upper Bound of Set/Real Numbers",
"Definition:Element",
"Definition:Element",
"Definition:Eleme... |
proofwiki-1441 | Complex Multiplication is Closed | The set of complex numbers $\C$ is closed under multiplication:
:$\forall z, w \in \C: z \times w \in \C$ | From the informal definition of complex numbers, we define the following:
:$z = x_1 + i y_1$
:$w = x_2 + i y_2$
where $i = \sqrt {-1}$ and $x_1, x_2, y_1, y_2 \in \R$.
Then from the definition of complex multiplication:
:$z w = \paren {x_1 x_2 - y_1 y_2} + i \paren {x_1 y_2 + x_2 y_1}$
From Real Numbers form Field:
:$x... | The [[Definition:Set|set]] of [[Definition:Complex Number|complex numbers]] $\C$ is [[Definition:Closed Algebraic Structure|closed]] under [[Definition:Complex Multiplication|multiplication]]:
:$\forall z, w \in \C: z \times w \in \C$ | From the informal definition of [[Definition:Complex Number/Definition 1|complex numbers]], we define the following:
:$z = x_1 + i y_1$
:$w = x_2 + i y_2$
where $i = \sqrt {-1}$ and $x_1, x_2, y_1, y_2 \in \R$.
Then from the definition of [[Definition:Complex Multiplication|complex multiplication]]:
:$z w = \paren {... | Complex Multiplication is Closed/Proof 1 | https://proofwiki.org/wiki/Complex_Multiplication_is_Closed | https://proofwiki.org/wiki/Complex_Multiplication_is_Closed/Proof_1 | [
"Complex Multiplication is Closed",
"Complex Multiplication",
"Algebraic Closure"
] | [
"Definition:Set",
"Definition:Complex Number",
"Definition:Closure (Abstract Algebra)/Algebraic Structure",
"Definition:Multiplication/Complex Numbers"
] | [
"Definition:Complex Number/Definition 1",
"Definition:Multiplication/Complex Numbers",
"Real Numbers form Field"
] |
proofwiki-1442 | Complex Multiplication is Closed | The set of complex numbers $\C$ is closed under multiplication:
:$\forall z, w \in \C: z \times w \in \C$ | From the formal definition of complex numbers, we define the following:
:$z = \tuple {x_1, y_1}$
:$w = \tuple {x_2, y_2}$
Then from the definition of complex multiplication:
:$z w = \tuple {x_1 x_2 - y_1 y_2, x_1 y_2 + x_2 y_1}$
From Real Numbers form Field:
:$x_1 x_2 - y_1 y_2 \in \R$
and:
:$x_1 y_2 + x_2 y_1 \in \R$
... | The [[Definition:Set|set]] of [[Definition:Complex Number|complex numbers]] $\C$ is [[Definition:Closed Algebraic Structure|closed]] under [[Definition:Complex Multiplication|multiplication]]:
:$\forall z, w \in \C: z \times w \in \C$ | From the formal definition of [[Definition:Complex Number#Formal Definition|complex numbers]], we define the following:
:$z = \tuple {x_1, y_1}$
:$w = \tuple {x_2, y_2}$
Then from the definition of [[Definition:Complex Multiplication|complex multiplication]]:
:$z w = \tuple {x_1 x_2 - y_1 y_2, x_1 y_2 + x_2 y_1}$
F... | Complex Multiplication is Closed/Proof 2 | https://proofwiki.org/wiki/Complex_Multiplication_is_Closed | https://proofwiki.org/wiki/Complex_Multiplication_is_Closed/Proof_2 | [
"Complex Multiplication is Closed",
"Complex Multiplication",
"Algebraic Closure"
] | [
"Definition:Set",
"Definition:Complex Number",
"Definition:Closure (Abstract Algebra)/Algebraic Structure",
"Definition:Multiplication/Complex Numbers"
] | [
"Definition:Complex Number",
"Definition:Multiplication/Complex Numbers",
"Real Numbers form Field"
] |
proofwiki-1443 | Complex Multiplication is Associative | The operation of multiplication on the set of complex numbers $\C$ is associative:
:$\forall z_1, z_2, z_3 \in \C: z_1 \paren {z_2 z_3} = \paren {z_1 z_2} z_3$ | From the definition of complex numbers, we define the following:
{{begin-eqn}}
{{eqn | l = z_1
| o = :=
| r = \tuple {x_1, y_1}
}}
{{eqn | l = z_2
| o = :=
| r = \tuple {x_2, y_2}
}}
{{eqn | l = z_3
| o = :=
| r = \tuple {x_3, y_3}
}}
{{end-eqn}}
where $x_1, x_2, x_3, y_1, y_2, y_3 \... | The operation of [[Definition:Complex Multiplication|multiplication]] on the [[Definition:Set|set]] of [[Definition:Complex Number|complex numbers]] $\C$ is [[Definition:Associative Operation|associative]]:
:$\forall z_1, z_2, z_3 \in \C: z_1 \paren {z_2 z_3} = \paren {z_1 z_2} z_3$ | From the definition of [[Definition:Complex Number/Definition 2|complex numbers]], we define the following:
{{begin-eqn}}
{{eqn | l = z_1
| o = :=
| r = \tuple {x_1, y_1}
}}
{{eqn | l = z_2
| o = :=
| r = \tuple {x_2, y_2}
}}
{{eqn | l = z_3
| o = :=
| r = \tuple {x_3, y_3}
}}
{{end... | Complex Multiplication is Associative | https://proofwiki.org/wiki/Complex_Multiplication_is_Associative | https://proofwiki.org/wiki/Complex_Multiplication_is_Associative | [
"Complex Multiplication is Associative",
"Complex Multiplication",
"Associative Law of Multiplication"
] | [
"Definition:Multiplication/Complex Numbers",
"Definition:Set",
"Definition:Complex Number",
"Definition:Associative Operation"
] | [
"Definition:Complex Number/Definition 2",
"Real Multiplication Distributes over Addition",
"Real Multiplication is Commutative",
"Real Multiplication Distributes over Addition"
] |
proofwiki-1444 | Multiple of Supremum | Let $S \subseteq \R: S \ne \O$ be a non-empty subset of the set of real numbers $\R$.
Let $S$ be bounded above.
Let $z \in \R: z > 0$ be a positive real number.
Then:
:$\ds \map {\sup_{x \mathop \in S} } {z x} = z \map {\sup_{x \mathop \in S} } x$ | Let $B = \map \sup S$.
Then by definition, $B$ is the smallest number such that $x \in S \implies x \le B$.
Let $T = \set {z x: x \in S}$.
Because $z > 0$, it follows that:
:$\forall x \in S: z x \le z B$
So $T$ is bounded above by $z B$.
By the Continuum Property, $T$ has a supremum which we will call $C$.
We need to ... | Let $S \subseteq \R: S \ne \O$ be a [[Definition:Non-Empty Set|non-empty]] [[Definition:Subset|subset]] of the [[Definition:Real Number|set of real numbers $\R$]].
Let $S$ be [[Definition:Bounded Above Set|bounded above]].
Let $z \in \R: z > 0$ be a [[Definition:Positive Real Number|positive real number]].
Then:
:$... | Let $B = \map \sup S$.
Then by definition, $B$ is the [[Definition:Smallest Element|smallest]] [[Definition:Real Number|number]] such that $x \in S \implies x \le B$.
Let $T = \set {z x: x \in S}$.
Because $z > 0$, it follows that:
:$\forall x \in S: z x \le z B$
So $T$ is [[Definition:Bounded Above Set|bounded abo... | Multiple of Supremum | https://proofwiki.org/wiki/Multiple_of_Supremum | https://proofwiki.org/wiki/Multiple_of_Supremum | [
"Real Analysis"
] | [
"Definition:Non-Empty Set",
"Definition:Subset",
"Definition:Real Number",
"Definition:Bounded Above Set",
"Definition:Positive/Real Number"
] | [
"Definition:Smallest Element",
"Definition:Real Number",
"Definition:Bounded Above Set",
"Continuum Property",
"Definition:Supremum of Set",
"Definition:Upper Bound of Set",
"Definition:Upper Bound of Set",
"Definition:Real Number",
"Definition:Smallest Element",
"Definition:Real Number",
"Defin... |
proofwiki-1445 | Cauchy's Mean Theorem | Let $x_1, x_2, \ldots, x_n \in \R$ be real numbers which are all positive.
Let $A_n$ be the arithmetic mean of $x_1, x_2, \ldots, x_n$.
Let $G_n$ be the geometric mean of $x_1, x_2, \ldots, x_n$.
Then:
:$A_n \ge G_n$
with equality holding {{iff}}:
:$\forall i, j \in \set {1, 2, \ldots, n}: x_i = x_j$
That is, {{iff}} a... | The arithmetic mean of $x_1, x_2, \ldots, x_n$ is defined as:
:$\ds A_n = \frac 1 n \paren {\sum_{k \mathop = 1}^n x_k}$
The geometric mean of $x_1, x_2, \ldots, x_n$ is defined as:
:$\ds G_n = \paren {\prod_{k \mathop = 1}^n x_k}^{1/n}$
We prove the result by induction:
For all $n \in \Z_{>0}$, let $\map P n$ be the p... | Let $x_1, x_2, \ldots, x_n \in \R$ be [[Definition:Real Number|real numbers]] which are all [[Definition:Positive Real Number|positive]].
Let $A_n$ be the [[Definition:Arithmetic Mean|arithmetic mean]] of $x_1, x_2, \ldots, x_n$.
Let $G_n$ be the [[Definition:Geometric Mean|geometric mean]] of $x_1, x_2, \ldots, x_n$... | The [[Definition:Arithmetic Mean|arithmetic mean]] of $x_1, x_2, \ldots, x_n$ is defined as:
:$\ds A_n = \frac 1 n \paren {\sum_{k \mathop = 1}^n x_k}$
The [[Definition:Geometric Mean|geometric mean]] of $x_1, x_2, \ldots, x_n$ is defined as:
:$\ds G_n = \paren {\prod_{k \mathop = 1}^n x_k}^{1/n}$
We prove the re... | Cauchy's Mean Theorem/Proof 1 | https://proofwiki.org/wiki/Cauchy's_Mean_Theorem | https://proofwiki.org/wiki/Cauchy's_Mean_Theorem/Proof_1 | [
"Cauchy's Mean Theorem",
"Arithmetic Mean",
"Geometric Mean",
"Inequalities"
] | [
"Definition:Real Number",
"Definition:Positive/Real Number",
"Definition:Arithmetic Mean",
"Definition:Geometric Mean",
"Definition:Term of Sequence"
] | [
"Definition:Arithmetic Mean",
"Definition:Geometric Mean",
"Principle of Mathematical Induction",
"Definition:Proposition",
"Definition:Positive/Real Number",
"Definition:Square Root",
"Definition:Basis for the Induction",
"Forward-Backward Induction",
"Definition:Induction Hypothesis",
"Definitio... |
proofwiki-1446 | Cauchy's Mean Theorem | Let $x_1, x_2, \ldots, x_n \in \R$ be real numbers which are all positive.
Let $A_n$ be the arithmetic mean of $x_1, x_2, \ldots, x_n$.
Let $G_n$ be the geometric mean of $x_1, x_2, \ldots, x_n$.
Then:
:$A_n \ge G_n$
with equality holding {{iff}}:
:$\forall i, j \in \set {1, 2, \ldots, n}: x_i = x_j$
That is, {{iff}} a... | Let:
:$\map f x = \ln x$
for $x > 0$.
With a view to apply Jensen's Inequality: Real Analysis: Corollary, we can show that $f$ is concave on $\openint 0 \infty$.
By Second Derivative of Concave Real Function is Non-Positive, it is sufficient to show that $\map {f' '} x \le 0$ for all $x > 0$.
We have, by Derivative o... | Let $x_1, x_2, \ldots, x_n \in \R$ be [[Definition:Real Number|real numbers]] which are all [[Definition:Positive Real Number|positive]].
Let $A_n$ be the [[Definition:Arithmetic Mean|arithmetic mean]] of $x_1, x_2, \ldots, x_n$.
Let $G_n$ be the [[Definition:Geometric Mean|geometric mean]] of $x_1, x_2, \ldots, x_n$... | Let:
:$\map f x = \ln x$
for $x > 0$.
With a view to apply [[Jensen's Inequality (Real Analysis)/Corollary|Jensen's Inequality: Real Analysis: Corollary]], we can show that $f$ is [[Definition:Concave Real Function|concave]] on $\openint 0 \infty$.
By [[Second Derivative of Concave Real Function is Non-Positive]],... | Cauchy's Mean Theorem/Proof 2 | https://proofwiki.org/wiki/Cauchy's_Mean_Theorem | https://proofwiki.org/wiki/Cauchy's_Mean_Theorem/Proof_2 | [
"Cauchy's Mean Theorem",
"Arithmetic Mean",
"Geometric Mean",
"Inequalities"
] | [
"Definition:Real Number",
"Definition:Positive/Real Number",
"Definition:Arithmetic Mean",
"Definition:Geometric Mean",
"Definition:Term of Sequence"
] | [
"Jensen's Inequality (Real Analysis)/Corollary",
"Definition:Concave Real Function",
"Second Derivative of Concave Real Function is Non-Positive",
"Derivative of Natural Logarithm Function",
"Power Rule for Derivatives",
"Definition:Concave Real Function",
"Definition:Positive/Real Number",
"Jensen's ... |
proofwiki-1447 | Cauchy's Mean Theorem | Let $x_1, x_2, \ldots, x_n \in \R$ be real numbers which are all positive.
Let $A_n$ be the arithmetic mean of $x_1, x_2, \ldots, x_n$.
Let $G_n$ be the geometric mean of $x_1, x_2, \ldots, x_n$.
Then:
:$A_n \ge G_n$
with equality holding {{iff}}:
:$\forall i, j \in \set {1, 2, \ldots, n}: x_i = x_j$
That is, {{iff}} a... | === Necessary Condition ===
Let:
:$\forall i, j \in \set {1, 2, \ldots, n}: x_i = x_j = x$
Then:
{{begin-eqn}}
{{eqn | l = A_n
| r = \dfrac 1 n \sum_{j \mathop = 1}^n x
| c =
}}
{{eqn | r = \dfrac 1 n n x
| c =
}}
{{eqn | r = x
| c =
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = G_n
| r = \p... | Let $x_1, x_2, \ldots, x_n \in \R$ be [[Definition:Real Number|real numbers]] which are all [[Definition:Positive Real Number|positive]].
Let $A_n$ be the [[Definition:Arithmetic Mean|arithmetic mean]] of $x_1, x_2, \ldots, x_n$.
Let $G_n$ be the [[Definition:Geometric Mean|geometric mean]] of $x_1, x_2, \ldots, x_n$... | === Necessary Condition ===
Let:
:$\forall i, j \in \set {1, 2, \ldots, n}: x_i = x_j = x$
Then:
{{begin-eqn}}
{{eqn | l = A_n
| r = \dfrac 1 n \sum_{j \mathop = 1}^n x
| c =
}}
{{eqn | r = \dfrac 1 n n x
| c =
}}
{{eqn | r = x
| c =
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = G_n
| ... | Cauchy's Mean Theorem/Proof of Equality Condition | https://proofwiki.org/wiki/Cauchy's_Mean_Theorem | https://proofwiki.org/wiki/Cauchy's_Mean_Theorem/Proof_of_Equality_Condition | [
"Cauchy's Mean Theorem",
"Arithmetic Mean",
"Geometric Mean",
"Inequalities"
] | [
"Definition:Real Number",
"Definition:Positive/Real Number",
"Definition:Arithmetic Mean",
"Definition:Geometric Mean",
"Definition:Term of Sequence"
] | [
"Principle of Mathematical Induction",
"Definition:Proposition",
"Definition:Basis for the Induction",
"Forward-Backward Induction",
"Definition:Induction Hypothesis",
"Cauchy's Mean Theorem/Proof of Equality Condition",
"Definition:Induction Step",
"Cauchy's Mean Theorem/Proof of Equality Condition",... |
proofwiki-1448 | Distance on Real Numbers is Metric | Let $x, y \in \R$ be real numbers.
Let $\map d {x, y}$ be the distance between $x$ and $y$:
:$\map d {x, y} = \size {x - y}$
Then $\map d {x, y}$ is a metric on $\R$.
Thus it follows that $\tuple {\R, d}$ is a metric space. | We check the metric space axioms in turn. | Let $x, y \in \R$ be [[Definition:Real Number|real numbers]].
Let $\map d {x, y}$ be the [[Definition:Distance between Real Numbers|distance]] between $x$ and $y$:
:$\map d {x, y} = \size {x - y}$
Then $\map d {x, y}$ is a [[Definition:Metric|metric]] on $\R$.
Thus it follows that $\tuple {\R, d}$ is a [[Definitio... | We check the [[Axiom:Metric Space Axioms|metric space axioms]] in turn. | Distance on Real Numbers is Metric | https://proofwiki.org/wiki/Distance_on_Real_Numbers_is_Metric | https://proofwiki.org/wiki/Distance_on_Real_Numbers_is_Metric | [
"Real Analysis",
"Real Number Line with Euclidean Metric"
] | [
"Definition:Real Number",
"Definition:Distance/Points/Real Numbers",
"Definition:Metric Space/Metric",
"Definition:Metric Space"
] | [
"Axiom:Metric Space Axioms"
] |
proofwiki-1449 | Convergent Sequence in Metric Space is Bounded | Let $M = \struct {A, d}$ be a metric space.
Let $\sequence {x_n}$ be a sequence in $M$ which is convergent, and so $x_n \to l$ as $n \to \infty$.
Then $\sequence {x_n}$ is bounded. | Let $M = \struct {A, d}$ be a metric space.
Let $\sequence {x_n}$ be a sequence in $M$ which is convergent, and so $x_n \to l$ as $n \to \infty$.
From the definition, in order to prove boundedness, all we need to do is find $K \in \R$ such that $\forall n \in \N: \map d {x_n, l} \le K$.
Since $\sequence {x_n}$ converge... | Let $M = \struct {A, d}$ be a [[Definition:Metric Space|metric space]].
Let $\sequence {x_n}$ be a [[Definition:Sequence|sequence in $M$]] which is [[Definition:Convergent Sequence (Metric Space)|convergent]], and so $x_n \to l$ as $n \to \infty$.
Then $\sequence {x_n}$ is [[Definition:Bounded Sequence|bounded]]. | Let $M = \struct {A, d}$ be a [[Definition:Metric Space|metric space]].
Let $\sequence {x_n}$ be a [[Definition:Sequence|sequence in $M$]] which is [[Definition:Convergent Sequence (Metric Space)|convergent]], and so $x_n \to l$ as $n \to \infty$.
From the definition, in order to prove [[Definition:Bounded Sequence|b... | Convergent Sequence in Metric Space is Bounded | https://proofwiki.org/wiki/Convergent_Sequence_in_Metric_Space_is_Bounded | https://proofwiki.org/wiki/Convergent_Sequence_in_Metric_Space_is_Bounded | [
"Limits of Sequences",
"Metric Spaces"
] | [
"Definition:Metric Space",
"Definition:Sequence",
"Definition:Convergent Sequence/Metric Space",
"Definition:Bounded Sequence"
] | [
"Definition:Metric Space",
"Definition:Sequence",
"Definition:Convergent Sequence/Metric Space",
"Definition:Bounded Sequence",
"Definition:Convergent Sequence/Metric Space"
] |
proofwiki-1450 | Convergent Sequence Minus Limit | Let $X$ be one of the standard number fields $\Q, \R, \C$.
Let $\sequence {x_n}$ be a sequence in $X$ which converges to $l$.
That is:
: $\ds \lim_{n \mathop \to \infty} x_n = l$
Then:
: $\ds \lim_{n \mathop \to \infty} \cmod {x_n - l} = 0$ | Let $\epsilon > 0$.
We need to show that there exists $N$ such that:
:$\forall n > N: \size {\paren {\size {x_n - l} - 0} } < \epsilon$
But:
:$\size {\paren {\size {x_n - l} - 0} } = \size {x_n - l}$
So what needs to be shown is just:
:$x_n \to l$ as $n \to \infty$
which is the definition of $\ds \lim_{n \mathop \to \i... | Let $X$ be one of the [[Definition:Standard Number Field|standard number fields]] $\Q, \R, \C$.
Let $\sequence {x_n}$ be a [[Definition:Sequence|sequence]] in $X$ which [[Definition:Convergent Sequence|converges]] to $l$.
That is:
: $\ds \lim_{n \mathop \to \infty} x_n = l$
Then:
: $\ds \lim_{n \mathop \to \infty} ... | Let $\epsilon > 0$.
We need to show that there exists $N$ such that:
:$\forall n > N: \size {\paren {\size {x_n - l} - 0} } < \epsilon$
But:
:$\size {\paren {\size {x_n - l} - 0} } = \size {x_n - l}$
So what needs to be shown is just:
:$x_n \to l$ as $n \to \infty$
which is the definition of $\ds \lim_{n \mathop \to... | Convergent Sequence Minus Limit/Proof 1 | https://proofwiki.org/wiki/Convergent_Sequence_Minus_Limit | https://proofwiki.org/wiki/Convergent_Sequence_Minus_Limit/Proof_1 | [
"Limits of Sequences",
"Convergent Sequence Minus Limit"
] | [
"Definition:Standard Number Field",
"Definition:Sequence",
"Definition:Convergent Sequence"
] | [] |
proofwiki-1451 | Convergent Sequence Minus Limit | Let $X$ be one of the standard number fields $\Q, \R, \C$.
Let $\sequence {x_n}$ be a sequence in $X$ which converges to $l$.
That is:
: $\ds \lim_{n \mathop \to \infty} x_n = l$
Then:
: $\ds \lim_{n \mathop \to \infty} \cmod {x_n - l} = 0$ | We note that all of $\Q, \R, \C$ can be considered as metric spaces.
Then under the usual metric:
: $\map d {x_n, l} = \cmod {x_n - l}$.
The result follows from the definition of metric:
:$\map d {x_n, l} = 0 \iff x_n = l$.
{{qed}} | Let $X$ be one of the [[Definition:Standard Number Field|standard number fields]] $\Q, \R, \C$.
Let $\sequence {x_n}$ be a [[Definition:Sequence|sequence]] in $X$ which [[Definition:Convergent Sequence|converges]] to $l$.
That is:
: $\ds \lim_{n \mathop \to \infty} x_n = l$
Then:
: $\ds \lim_{n \mathop \to \infty} ... | We note that all of $\Q, \R, \C$ can be considered as [[Definition:Euclidean Space|metric spaces]].
Then under the [[Definition:Usual Metric|usual metric]]:
: $\map d {x_n, l} = \cmod {x_n - l}$.
The result follows from the definition of [[Definition:Metric|metric]]:
:$\map d {x_n, l} = 0 \iff x_n = l$.
{{qed}} | Convergent Sequence Minus Limit/Proof 2 | https://proofwiki.org/wiki/Convergent_Sequence_Minus_Limit | https://proofwiki.org/wiki/Convergent_Sequence_Minus_Limit/Proof_2 | [
"Limits of Sequences",
"Convergent Sequence Minus Limit"
] | [
"Definition:Standard Number Field",
"Definition:Sequence",
"Definition:Convergent Sequence"
] | [
"Definition:Euclidean Space",
"Definition:Usual Metric",
"Definition:Metric Space/Metric"
] |
proofwiki-1452 | One Plus Reciprocal to the Nth | Let $\sequence {x_n}$ be the sequence in $\R$ defined as $x_n = \paren {1 + \dfrac 1 n}^n$.
Then $\sequence {x_n}$ converges to a limit as $n$ increases without bound. | First we show that $\sequence {x_n}$ is increasing.
Let $a_1 = a_2 = \cdots = a_{n - 1} = 1 + \dfrac 1 {n - 1}$.
Let $a_n = 1$.
Let:
:$A_n$ be the arithmetic mean of $a_1 \ldots a_n$
:$G_n$ be the geometric mean of $a_1 \ldots a_n$
Thus:
:$A_n = \dfrac {\paren {n - 1} \paren {1 + \dfrac 1 {n - 1} } + 1} n = \dfrac {n +... | Let $\sequence {x_n}$ be the [[Definition:Real Sequence|sequence in $\R$]] defined as $x_n = \paren {1 + \dfrac 1 n}^n$.
Then $\sequence {x_n}$ [[Definition:Convergent Real Sequence|converges]] to a [[Definition:Limit of Real Sequence|limit]] as $n$ [[Definition:Increase Without Bound|increases without bound]]. | First we show that $\sequence {x_n}$ is [[Definition:Increasing Real Sequence|increasing]].
Let $a_1 = a_2 = \cdots = a_{n - 1} = 1 + \dfrac 1 {n - 1}$.
Let $a_n = 1$.
Let:
:$A_n$ be the [[Definition:Arithmetic Mean|arithmetic mean]] of $a_1 \ldots a_n$
:$G_n$ be the [[Definition:Geometric Mean|geometric mean]] of $... | One Plus Reciprocal to the Nth | https://proofwiki.org/wiki/One_Plus_Reciprocal_to_the_Nth | https://proofwiki.org/wiki/One_Plus_Reciprocal_to_the_Nth | [
"Limits of Sequences",
"Reciprocals"
] | [
"Definition:Real Sequence",
"Definition:Convergent Sequence/Real Numbers",
"Definition:Limit of Sequence/Real Numbers",
"Definition:Bounded Above Mapping/Unbounded"
] | [
"Definition:Increasing/Sequence/Real Sequence",
"Definition:Arithmetic Mean",
"Definition:Geometric Mean",
"Cauchy's Mean Theorem",
"Definition:Increasing/Sequence/Real Sequence",
"Definition:Bounded Above Sequence/Real",
"Binomial Theorem",
"Definition:Bounded Above Sequence/Real",
"Monotone Conver... |
proofwiki-1453 | Between two Real Numbers exists Rational Number | Let $a, b \in \R$ be real numbers such that $a < b$.
Then:
:$\exists r \in \Q: a < r < b$ | Suppose that $a \ge 0$.
As $a < b$ it follows that $a \ne b$ and so $b - a \ne 0$.
Thus:
:$\dfrac 1 {b - a} \in \R$
By the Axiom of Archimedes:
:$\exists n \in \N: n > \dfrac 1 {b - a}$
Let $M := \set {x \in \N: \dfrac x n > a}$.
By the Well-Ordering Principle, there exists $m \in \N$ such that $m$ is the smallest elem... | Let $a, b \in \R$ be [[Definition:Real Number|real numbers]] such that $a < b$.
Then:
:$\exists r \in \Q: a < r < b$ | Suppose that $a \ge 0$.
As $a < b$ it follows that $a \ne b$ and so $b - a \ne 0$.
Thus:
:$\dfrac 1 {b - a} \in \R$
By the [[Axiom of Archimedes]]:
:$\exists n \in \N: n > \dfrac 1 {b - a}$
Let $M := \set {x \in \N: \dfrac x n > a}$.
By the [[Well-Ordering Principle]], there exists $m \in \N$ such that $m$ is the ... | Between two Real Numbers exists Rational Number/Proof 1 | https://proofwiki.org/wiki/Between_two_Real_Numbers_exists_Rational_Number | https://proofwiki.org/wiki/Between_two_Real_Numbers_exists_Rational_Number/Proof_1 | [
"Real Analysis",
"Between two Real Numbers exists Rational Number"
] | [
"Definition:Real Number"
] | [
"Axiom of Archimedes",
"Well-Ordering Principle",
"Definition:Smallest Element",
"Definition:Smallest Element",
"Ordering of Reciprocals",
"Definition:Rational Number"
] |
proofwiki-1454 | Between two Real Numbers exists Rational Number | Let $a, b \in \R$ be real numbers such that $a < b$.
Then:
:$\exists r \in \Q: a < r < b$ | As $a < b$ it follows that $a \ne b$ and so $b - a \ne 0$.
Thus:
:$\dfrac 1 {b - a} \in \R$
By the Axiom of Archimedes:
:$\exists n \in \N: n > \dfrac 1 {b - a}$
Let $M := \set {x \in \Z: x > a n}$.
By Set of Integers Bounded Below has Smallest Element, there exists $m \in \Z$ such that $m$ is the smallest element of $... | Let $a, b \in \R$ be [[Definition:Real Number|real numbers]] such that $a < b$.
Then:
:$\exists r \in \Q: a < r < b$ | As $a < b$ it follows that $a \ne b$ and so $b - a \ne 0$.
Thus:
:$\dfrac 1 {b - a} \in \R$
By the [[Axiom of Archimedes]]:
:$\exists n \in \N: n > \dfrac 1 {b - a}$
Let $M := \set {x \in \Z: x > a n}$.
By [[Set of Integers Bounded Below has Smallest Element]], there exists $m \in \Z$ such that $m$ is the [[Definit... | Between two Real Numbers exists Rational Number/Proof 2 | https://proofwiki.org/wiki/Between_two_Real_Numbers_exists_Rational_Number | https://proofwiki.org/wiki/Between_two_Real_Numbers_exists_Rational_Number/Proof_2 | [
"Real Analysis",
"Between two Real Numbers exists Rational Number"
] | [
"Definition:Real Number"
] | [
"Axiom of Archimedes",
"Set of Integers Bounded Below has Smallest Element",
"Definition:Smallest Element",
"Definition:Smallest Element",
"Ordering of Reciprocals"
] |
proofwiki-1455 | Power over Factorial | Let $x \in \R: x > 0$ be a positive real number.
Let $\sequence {x_n}$ be the sequence in $\R$ defined as $x_n = \dfrac {x^n} {n!}$.
Then $\sequence {x_n}$ converges to zero. | We need to show that $x_n \to 0$ as $n \to \infty$.
Let $N \in \N$ be the smallest natural number which satisfies $N > x$.
(From the Axiom of Archimedes, such an $N$ always exists.)
First we show that:
: $\forall n > N: \dfrac {x^n} {n!} \le \dfrac {x^{N - 1} } {\paren {N - 1}!} \paren {\dfrac x N}^{n - N + 1}$
Note th... | Let $x \in \R: x > 0$ be a positive [[Definition:Real Number|real number]].
Let $\sequence {x_n}$ be the [[Definition:Real Sequence|sequence in $\R$]] defined as $x_n = \dfrac {x^n} {n!}$.
Then $\sequence {x_n}$ [[Definition:Convergent Sequence|converges]] to zero. | We need to show that $x_n \to 0$ as $n \to \infty$.
Let $N \in \N$ be the smallest [[Definition:Natural Numbers|natural number]] which satisfies $N > x$.
(From the [[Axiom of Archimedes]], such an $N$ always exists.)
First we show that:
: $\forall n > N: \dfrac {x^n} {n!} \le \dfrac {x^{N - 1} } {\paren {N - 1}!} \... | Power over Factorial | https://proofwiki.org/wiki/Power_over_Factorial | https://proofwiki.org/wiki/Power_over_Factorial | [
"Limits of Sequences"
] | [
"Definition:Real Number",
"Definition:Real Sequence",
"Definition:Convergent Sequence"
] | [
"Definition:Natural Numbers",
"Axiom of Archimedes",
"Sequence of Powers of Number less than One",
"Combination Theorem for Sequences/Real/Multiple Rule",
"Squeeze Theorem/Sequences/Real Numbers"
] |
proofwiki-1456 | Sequence of Powers of Reciprocals is Null Sequence | Let $r \in \Q_{>0}$ be a strictly positive rational number.
Let $\sequence {x_n}$ be the sequence in $\R$ defined as:
:$x_n = \dfrac 1 {n^r}$
Then $\sequence {x_n}$ is a null sequence. | Let $\epsilon \in \R_{>0}$.
We need to show that:
:$\exists N \in \N: n > N \implies \size {\dfrac 1 {n^r} } < \epsilon$
That is, that $n^r > 1 / \epsilon$.
Let us choose $N = \ceiling {\paren {1 / \epsilon}^{1/r} }$.
By Reciprocal of Strictly Positive Real Number is Strictly Positive and power of positive real number ... | Let $r \in \Q_{>0}$ be a [[Definition:Strictly Positive Rational Number|strictly positive rational number]].
Let $\sequence {x_n}$ be the [[Definition:Real Sequence|sequence in $\R$]] defined as:
:$x_n = \dfrac 1 {n^r}$
Then $\sequence {x_n}$ is a [[Definition:Null Sequence (Analysis)|null sequence]]. | Let $\epsilon \in \R_{>0}$.
We need to show that:
:$\exists N \in \N: n > N \implies \size {\dfrac 1 {n^r} } < \epsilon$
That is, that $n^r > 1 / \epsilon$.
Let us choose $N = \ceiling {\paren {1 / \epsilon}^{1/r} }$.
By [[Reciprocal of Strictly Positive Real Number is Strictly Positive]] and [[Power of Positive R... | Sequence of Powers of Reciprocals is Null Sequence | https://proofwiki.org/wiki/Sequence_of_Powers_of_Reciprocals_is_Null_Sequence | https://proofwiki.org/wiki/Sequence_of_Powers_of_Reciprocals_is_Null_Sequence | [
"Limits of Sequences",
"Reciprocals",
"Sequence of Powers of Reciprocals is Null Sequence"
] | [
"Definition:Strictly Positive/Rational Number",
"Definition:Real Sequence",
"Definition:Null Sequence/Analysis"
] | [
"Reciprocal of Strictly Positive Real Number is Strictly Positive",
"Power of Positive Real Number is Positive/Rational Number",
"Positive Power Function on Non-negative Reals is Strictly Increasing"
] |
proofwiki-1457 | Euler's Number: Limit of Sequence implies Limit of Series | Let Euler's number $e$ be defined as:
:$\ds e := \lim_{n \mathop \to \infty} \paren {1 + \frac 1 n}^n$
Then:
:$\ds e = \sum_{k \mathop = 0}^\infty \frac 1 {k!}$
That is:
:$e = \dfrac 1 {0!} + \dfrac 1 {1!} + \dfrac 1 {2!} + \dfrac 1 {3!} + \dfrac 1 {4!} \cdots$ | We expand $\paren {1 + \dfrac 1 n}^n$ by the Binomial Theorem, using that $\dfrac {n - k} n = 1 - \dfrac k n$:
{{begin-eqn}}
{{eqn | l = \paren {1 + \frac 1 n}^n
| r = 1 + n \paren {\frac 1 n} + \frac {n \paren {n - 1} } 2 \paren {\frac 1 n}^2 + \cdots + \paren {\frac 1 n}^n
| c =
}}
{{eqn | r = \frac 1 {0!... | Let [[Definition:Euler's Number as Limit of Sequence|Euler's number $e$]] be defined as:
:$\ds e := \lim_{n \mathop \to \infty} \paren {1 + \frac 1 n}^n$
Then:
:$\ds e = \sum_{k \mathop = 0}^\infty \frac 1 {k!}$
That is:
:$e = \dfrac 1 {0!} + \dfrac 1 {1!} + \dfrac 1 {2!} + \dfrac 1 {3!} + \dfrac 1 {4!} \cdots$ | We expand $\paren {1 + \dfrac 1 n}^n$ by the [[Binomial Theorem]], using that $\dfrac {n - k} n = 1 - \dfrac k n$:
{{begin-eqn}}
{{eqn | l = \paren {1 + \frac 1 n}^n
| r = 1 + n \paren {\frac 1 n} + \frac {n \paren {n - 1} } 2 \paren {\frac 1 n}^2 + \cdots + \paren {\frac 1 n}^n
| c =
}}
{{eqn | r = \frac ... | Euler's Number: Limit of Sequence implies Limit of Series/Proof 1 | https://proofwiki.org/wiki/Euler's_Number:_Limit_of_Sequence_implies_Limit_of_Series | https://proofwiki.org/wiki/Euler's_Number:_Limit_of_Sequence_implies_Limit_of_Series/Proof_1 | [
"Euler's Number"
] | [
"Definition:Euler's Number/Limit of Sequence"
] | [
"Binomial Theorem",
"Sequence of Powers of Reciprocals is Null Sequence",
"Combination Theorem for Sequences"
] |
proofwiki-1458 | Euler's Number: Limit of Sequence implies Limit of Series | Let Euler's number $e$ be defined as:
:$\ds e := \lim_{n \mathop \to \infty} \paren {1 + \frac 1 n}^n$
Then:
:$\ds e = \sum_{k \mathop = 0}^\infty \frac 1 {k!}$
That is:
:$e = \dfrac 1 {0!} + \dfrac 1 {1!} + \dfrac 1 {2!} + \dfrac 1 {3!} + \dfrac 1 {4!} \cdots$ | It will be shown that:
:$\ds \lim_{n \mathop \to \infty} \paren {1 + \frac 1 n}^n = \sum_{k \mathop = 0}^\infty \frac 1 {k!}$
Let $t_n := \paren {1 + \dfrac 1 n}^n$
Then:
:$t_n = \dfrac 1 {0!} + \dfrac 1 {1!} + \paren {1 - \dfrac 1 n} \dfrac 1 {2!} + \paren {1 - \dfrac 1 n} \paren {1 - \dfrac 2 n} \dfrac 1 {3!} + \cdot... | Let [[Definition:Euler's Number as Limit of Sequence|Euler's number $e$]] be defined as:
:$\ds e := \lim_{n \mathop \to \infty} \paren {1 + \frac 1 n}^n$
Then:
:$\ds e = \sum_{k \mathop = 0}^\infty \frac 1 {k!}$
That is:
:$e = \dfrac 1 {0!} + \dfrac 1 {1!} + \dfrac 1 {2!} + \dfrac 1 {3!} + \dfrac 1 {4!} \cdots$ | It will be shown that:
:$\ds \lim_{n \mathop \to \infty} \paren {1 + \frac 1 n}^n = \sum_{k \mathop = 0}^\infty \frac 1 {k!}$
Let $t_n := \paren {1 + \dfrac 1 n}^n$
Then:
:$t_n = \dfrac 1 {0!} + \dfrac 1 {1!} + \paren {1 - \dfrac 1 n} \dfrac 1 {2!} + \paren {1 - \dfrac 1 n} \paren {1 - \dfrac 2 n} \dfrac 1 {3!} + \c... | Euler's Number: Limit of Sequence implies Limit of Series/Proof 2 | https://proofwiki.org/wiki/Euler's_Number:_Limit_of_Sequence_implies_Limit_of_Series | https://proofwiki.org/wiki/Euler's_Number:_Limit_of_Sequence_implies_Limit_of_Series/Proof_2 | [
"Euler's Number"
] | [
"Definition:Euler's Number/Limit of Sequence"
] | [] |
proofwiki-1459 | Degree Equation | Let $E$, $K$ and $F$ be fields.
Let $E / K$ and $K / F$ be finite field extensions.
Then:
:$E / F$ is a finite field extension
:$\index E F = \index E K \index K F$
where $\index E F$ denotes the degree of $E / F$ | First, note that $E / F$ is a field extension as $F \subseteq K \subseteq E$.
Suppose that $\index E K = m$ and $\index K F = n$.
Let $\alpha = \set {a_1, \ldots, a_m}$ be a basis of $E / K$, and $\beta = \set {b_1, \ldots, b_n}$ be a basis of $K / F$.
We wish to prove that the set:
:$\gamma = \set {a_i b_j: 1 \le i \l... | Let $E$, $K$ and $F$ be [[Definition:Field (Abstract Algebra)|fields]].
Let $E / K$ and $K / F$ be [[Definition:Finite Field Extension|finite field extensions]].
Then:
:$E / F$ is a [[Definition:Finite Field Extension|finite field extension]]
:$\index E F = \index E K \index K F$
where $\index E F$ denotes the [[De... | First, note that $E / F$ is a [[Definition:Field Extension|field extension]] as $F \subseteq K \subseteq E$.
Suppose that $\index E K = m$ and $\index K F = n$.
Let $\alpha = \set {a_1, \ldots, a_m}$ be a [[Definition:Basis (Linear Algebra)|basis]] of $E / K$, and $\beta = \set {b_1, \ldots, b_n}$ be a [[Definition:B... | Degree Equation | https://proofwiki.org/wiki/Degree_Equation | https://proofwiki.org/wiki/Degree_Equation | [
"Degree Equation",
"Degrees of Field Extensions"
] | [
"Definition:Field (Abstract Algebra)",
"Definition:Field Extension/Degree/Finite",
"Definition:Field Extension/Degree/Finite",
"Definition:Field Extension/Degree"
] | [
"Definition:Field Extension",
"Definition:Basis (Linear Algebra)",
"Definition:Basis of Vector Space",
"Definition:Set",
"Definition:Basis of Vector Space",
"Definition:Basis of Vector Space",
"Definition:Linearly Independent/Set",
"Definition:Generator of Vector Space",
"Definition:Linearly Indepen... |
proofwiki-1460 | Convergent Sequence in Metric Space has Unique Limit | Let $M = \struct {A, d}$ be a metric space.
Let $\sequence {x_n}$ be a sequence in $M$.
Then $\sequence {x_n}$ can have at most one limit in $M$. | Suppose $\ds \lim_{n \mathop \to \infty} x_n = l$ and $\ds \lim_{n \mathop \to \infty} x_n = m$.
Let $\epsilon > 0$.
Then, provided $n$ is sufficiently large:
{{begin-eqn}}
{{eqn | l = \map d {l, m}
| o = \le
| r = \map d {l, x_n} + \map d {x_n, m}
| c = Triangle Inequality
}}
{{eqn | o = <
| r ... | Let $M = \struct {A, d}$ be a [[Definition:Metric Space|metric space]].
Let $\sequence {x_n}$ be a [[Definition:Sequence|sequence]] in $M$.
Then $\sequence {x_n}$ can have at most one [[Definition:Limit of Sequence (Metric Space)|limit]] in $M$. | Suppose $\ds \lim_{n \mathop \to \infty} x_n = l$ and $\ds \lim_{n \mathop \to \infty} x_n = m$.
Let $\epsilon > 0$.
Then, provided $n$ is [[Definition:Sufficiently Large|sufficiently large]]:
{{begin-eqn}}
{{eqn | l = \map d {l, m}
| o = \le
| r = \map d {l, x_n} + \map d {x_n, m}
| c = [[Triangl... | Convergent Sequence in Metric Space has Unique Limit/Proof 1 | https://proofwiki.org/wiki/Convergent_Sequence_in_Metric_Space_has_Unique_Limit | https://proofwiki.org/wiki/Convergent_Sequence_in_Metric_Space_has_Unique_Limit/Proof_1 | [
"Limits of Sequences",
"Metric Spaces",
"Convergent Sequence in Metric Space has Unique Limit"
] | [
"Definition:Metric Space",
"Definition:Sequence",
"Definition:Limit of Sequence/Metric Space"
] | [
"Definition:Sufficiently Large",
"Triangle Inequality",
"Real Plus Epsilon"
] |
proofwiki-1461 | Convergent Sequence in Metric Space has Unique Limit | Let $M = \struct {A, d}$ be a metric space.
Let $\sequence {x_n}$ be a sequence in $M$.
Then $\sequence {x_n}$ can have at most one limit in $M$. | We have that a Metric Space is Hausdorff.
The result then follows from Convergent Sequence in Hausdorff Space has Unique Limit.
{{qed}} | Let $M = \struct {A, d}$ be a [[Definition:Metric Space|metric space]].
Let $\sequence {x_n}$ be a [[Definition:Sequence|sequence]] in $M$.
Then $\sequence {x_n}$ can have at most one [[Definition:Limit of Sequence (Metric Space)|limit]] in $M$. | We have that a [[Metric Space is Hausdorff]].
The result then follows from [[Convergent Sequence in Hausdorff Space has Unique Limit]].
{{qed}} | Convergent Sequence in Metric Space has Unique Limit/Proof 2 | https://proofwiki.org/wiki/Convergent_Sequence_in_Metric_Space_has_Unique_Limit | https://proofwiki.org/wiki/Convergent_Sequence_in_Metric_Space_has_Unique_Limit/Proof_2 | [
"Limits of Sequences",
"Metric Spaces",
"Convergent Sequence in Metric Space has Unique Limit"
] | [
"Definition:Metric Space",
"Definition:Sequence",
"Definition:Limit of Sequence/Metric Space"
] | [
"Metric Space is T2",
"Convergent Sequence in T2 Space has Unique Limit"
] |
proofwiki-1462 | Lower and Upper Bounds for Sequences | Let $\sequence {x_n}$ be a sequence in $\R$.
Let $x_n \to l$ as $n \to \infty$.
Then:
:$(1): \quad \forall n \in \N: x_n \ge a \implies l \ge a$
:$(2): \quad \forall n \in \N: x_n \le b \implies l \le b$ | $(1): \quad \forall n \in \N: x_n \ge a \implies l \ge a$:
Let $\epsilon > 0$.
Then:
:$\exists N \in \N: n > N \implies \size {x_n - l} < \epsilon$
So from Negative of Absolute Value:
:$l - \epsilon < x_n < l + \epsilon$
But $x_n \ge a$, so:
:$a \le x_n < l + \epsilon$
Thus, for ''any'' $\epsilon > 0$:
:$a < l + \epsil... | Let $\sequence {x_n}$ be a [[Definition:Real Sequence|sequence in $\R$]].
Let $x_n \to l$ as $n \to \infty$.
Then:
:$(1): \quad \forall n \in \N: x_n \ge a \implies l \ge a$
:$(2): \quad \forall n \in \N: x_n \le b \implies l \le b$ | $(1): \quad \forall n \in \N: x_n \ge a \implies l \ge a$:
Let $\epsilon > 0$.
Then:
:$\exists N \in \N: n > N \implies \size {x_n - l} < \epsilon$
So from [[Negative of Absolute Value]]:
:$l - \epsilon < x_n < l + \epsilon$
But $x_n \ge a$, so:
:$a \le x_n < l + \epsilon$
Thus, for ''any'' $\epsilon > 0$:
:$a < l... | Lower and Upper Bounds for Sequences | https://proofwiki.org/wiki/Lower_and_Upper_Bounds_for_Sequences | https://proofwiki.org/wiki/Lower_and_Upper_Bounds_for_Sequences | [
"Limits of Sequences"
] | [
"Definition:Real Sequence"
] | [
"Negative of Absolute Value",
"Real Plus Epsilon"
] |
proofwiki-1463 | Divergent Sequence may be Bounded | While every Convergent Sequence is Bounded, it does not follow that every bounded sequence is convergent.
That is, there exist bounded sequences which are divergent. | Let $\sequence {x_n}$ be the sequence in $\R$ which forms the basis of Grandi's series, defined as:
:$x_n = \paren {-1}^n$
It is clear that $\sequence {x_n}$ is bounded: above by $1$ and below by $-1$.
{{AimForCont}} $x_n \to l$ as $n \to \infty$.
Let $\epsilon > 0$.
Then $\exists N \in \R: \forall n > N: \size {\paren... | While every [[Convergent Sequence is Bounded]], it does not follow that every [[Definition:Bounded Sequence|bounded sequence]] is [[Definition:Convergent Sequence|convergent]].
That is, there exist [[Definition:Bounded Sequence|bounded sequences]] which are [[Definition:Divergent Sequence|divergent]]. | Let $\sequence {x_n}$ be the [[Definition:Real Sequence|sequence in $\R$]] which forms the basis of [[Definition:Grandi's Series|Grandi's series]], defined as:
:$x_n = \paren {-1}^n$
It is clear that $\sequence {x_n}$ is [[Definition:Bounded Real Sequence|bounded]]: [[Definition:Bounded Above Real Sequence|above]] by ... | Divergent Sequence may be Bounded/Proof 1 | https://proofwiki.org/wiki/Divergent_Sequence_may_be_Bounded | https://proofwiki.org/wiki/Divergent_Sequence_may_be_Bounded/Proof_1 | [
"Divergent Sequence may be Bounded",
"Limits of Sequences",
"Divergent Sequences"
] | [
"Convergent Sequence in Metric Space is Bounded",
"Definition:Bounded Sequence",
"Definition:Convergent Sequence",
"Definition:Bounded Sequence",
"Definition:Divergent Sequence"
] | [
"Definition:Real Sequence",
"Definition:Grandi's Series",
"Definition:Bounded Sequence/Real",
"Definition:Bounded Above Sequence/Real",
"Definition:Bounded Below Sequence/Real",
"Triangle Inequality/Real Numbers",
"Definition:Contradiction",
"Definition:Limit of Sequence/Real Numbers",
"Definition:B... |
proofwiki-1464 | Divergent Sequence may be Bounded | While every Convergent Sequence is Bounded, it does not follow that every bounded sequence is convergent.
That is, there exist bounded sequences which are divergent. | Let $\sequence {x_n}$ be the sequence in $\R$ which forms the basis of Grandi's series, defined as:
:$x_n = \paren {-1}^n$
It is clear that $\sequence {x_n}$ is bounded: above by $1$ and below by $-1$.
Note the following subsequences of $\sequence {x_n}$:
:$(1): \quad \sequence {x_{n_r} }$ where $\sequence {n_r}$ is t... | While every [[Convergent Sequence is Bounded]], it does not follow that every [[Definition:Bounded Sequence|bounded sequence]] is [[Definition:Convergent Sequence|convergent]].
That is, there exist [[Definition:Bounded Sequence|bounded sequences]] which are [[Definition:Divergent Sequence|divergent]]. | Let $\sequence {x_n}$ be the [[Definition:Real Sequence|sequence in $\R$]] which forms the basis of [[Definition:Grandi's Series|Grandi's series]], defined as:
:$x_n = \paren {-1}^n$
It is clear that $\sequence {x_n}$ is [[Definition:Bounded Real Sequence|bounded]]: [[Definition:Bounded Above Real Sequence|above]] by... | Divergent Sequence may be Bounded/Proof 2 | https://proofwiki.org/wiki/Divergent_Sequence_may_be_Bounded | https://proofwiki.org/wiki/Divergent_Sequence_may_be_Bounded/Proof_2 | [
"Divergent Sequence may be Bounded",
"Limits of Sequences",
"Divergent Sequences"
] | [
"Convergent Sequence in Metric Space is Bounded",
"Definition:Bounded Sequence",
"Definition:Convergent Sequence",
"Definition:Bounded Sequence",
"Definition:Divergent Sequence"
] | [
"Definition:Real Sequence",
"Definition:Grandi's Series",
"Definition:Bounded Sequence/Real",
"Definition:Bounded Above Sequence/Real",
"Definition:Bounded Below Sequence/Real",
"Definition:Subsequence",
"Definition:Integer Sequence",
"Definition:Integer Sequence",
"Definition:Real Sequence",
"Def... |
proofwiki-1465 | Ring Homomorphism from Division Ring is Monomorphism or Zero Homomorphism | Let $\struct {R, +_R, \circ}$ and $\struct {S, +_S, *}$ be rings whose zeros are $0_R$ and $0_S$ respectively.
Let $\phi: R \to S$ be a ring homomorphism.
If $R$ is a division ring, then either:
:$(1): \quad \phi$ is a monomorphism (that is, $\phi$ is injective)
:$(2): \quad \phi$ is the zero homomorphism (that is, $\f... | We have that:
:The kernel of a homomorphism is an ideal of $R$
:the only ideals of a division ring are trivial.
So $\map \ker \phi = \set {0_R}$ or $R$.
If $\map \ker \phi = \set {0_R}$, then $\phi$ is injective by Kernel is Trivial iff Monomorphism.
If $\map \ker \phi = R$, $\phi$ is the zero homomorphism by definit... | Let $\struct {R, +_R, \circ}$ and $\struct {S, +_S, *}$ be [[Definition:Ring (Abstract Algebra)|rings]] whose [[Definition:Ring Zero|zeros]] are $0_R$ and $0_S$ respectively.
Let $\phi: R \to S$ be a [[Definition:Ring Homomorphism|ring homomorphism]].
If $R$ is a [[Definition:Division Ring|division ring]], then eith... | We have that:
:The [[Kernel of Ring Epimorphism is Ideal|kernel of a homomorphism is an ideal of $R$]]
:[[Ideals of Division Ring|the only ideals of a division ring are trivial]].
So $\map \ker \phi = \set {0_R}$ or $R$.
If $\map \ker \phi = \set {0_R}$, then $\phi$ is [[Definition:Injection|injective]] by [[Kernel... | Ring Homomorphism from Division Ring is Monomorphism or Zero Homomorphism/Proof 1 | https://proofwiki.org/wiki/Ring_Homomorphism_from_Division_Ring_is_Monomorphism_or_Zero_Homomorphism | https://proofwiki.org/wiki/Ring_Homomorphism_from_Division_Ring_is_Monomorphism_or_Zero_Homomorphism/Proof_1 | [
"Ring Homomorphism from Division Ring is Monomorphism or Zero Homomorphism",
"Division Rings",
"Monomorphisms (Abstract Algebra)"
] | [
"Definition:Ring (Abstract Algebra)",
"Definition:Ring Zero",
"Definition:Ring Homomorphism",
"Definition:Division Ring",
"Definition:Ring Monomorphism",
"Definition:Injection",
"Definition:Zero Homomorphism"
] | [
"Kernel of Ring Epimorphism is Ideal",
"Ideals of Division Ring",
"Definition:Injection",
"Kernel is Trivial iff Monomorphism",
"Definition:Zero Homomorphism"
] |
proofwiki-1466 | Ring Homomorphism from Division Ring is Monomorphism or Zero Homomorphism | Let $\struct {R, +_R, \circ}$ and $\struct {S, +_S, *}$ be rings whose zeros are $0_R$ and $0_S$ respectively.
Let $\phi: R \to S$ be a ring homomorphism.
If $R$ is a division ring, then either:
:$(1): \quad \phi$ is a monomorphism (that is, $\phi$ is injective)
:$(2): \quad \phi$ is the zero homomorphism (that is, $\f... | From Surjection by Restriction of Codomain, we can restrict the codomain of $\phi$ and consider the mapping $\phi': R \to \Img R$
As $\phi'$ is now a surjective homomorphism, it is by definition an epimorphism.
Then an Epimorphism from Division Ring to Ring is either null or an isomorphism.
As an isomorphism is by defi... | Let $\struct {R, +_R, \circ}$ and $\struct {S, +_S, *}$ be [[Definition:Ring (Abstract Algebra)|rings]] whose [[Definition:Ring Zero|zeros]] are $0_R$ and $0_S$ respectively.
Let $\phi: R \to S$ be a [[Definition:Ring Homomorphism|ring homomorphism]].
If $R$ is a [[Definition:Division Ring|division ring]], then eith... | From [[Surjection by Restriction of Codomain]], we can restrict the [[Definition:Codomain of Mapping|codomain]] of $\phi$ and consider the mapping $\phi': R \to \Img R$
As $\phi'$ is now a [[Definition:Surjection|surjective]] [[Definition:Ring Homomorphism|homomorphism]], it is by definition an [[Definition:Ring Epimo... | Ring Homomorphism from Division Ring is Monomorphism or Zero Homomorphism/Proof 2 | https://proofwiki.org/wiki/Ring_Homomorphism_from_Division_Ring_is_Monomorphism_or_Zero_Homomorphism | https://proofwiki.org/wiki/Ring_Homomorphism_from_Division_Ring_is_Monomorphism_or_Zero_Homomorphism/Proof_2 | [
"Ring Homomorphism from Division Ring is Monomorphism or Zero Homomorphism",
"Division Rings",
"Monomorphisms (Abstract Algebra)"
] | [
"Definition:Ring (Abstract Algebra)",
"Definition:Ring Zero",
"Definition:Ring Homomorphism",
"Definition:Division Ring",
"Definition:Ring Monomorphism",
"Definition:Injection",
"Definition:Zero Homomorphism"
] | [
"Restriction of Mapping to Image is Surjection",
"Definition:Codomain (Set Theory)/Mapping",
"Definition:Surjection",
"Definition:Ring Homomorphism",
"Definition:Ring Epimorphism",
"Epimorphism from Division Ring to Ring",
"Definition:Null Ring",
"Definition:Isomorphism (Abstract Algebra)",
"Definit... |
proofwiki-1467 | Reciprocal of Null Sequence | Let $\sequence {x_n}$ be a sequence in $\R$.
Let $\forall n \in \N: x_n > 0$.
Then:
:$x_n \to 0$ as $n \to \infty$ {{iff}} $\size {\dfrac 1 {x_n} } \to \infty$ as $n \to \infty$ | Suppose $x_n \to 0$ as $n \to \infty$.
Let $H > 0$.
So $H^{-1} > 0$.
Since $x_n \to 0$ as $n \to \infty$:
:$\exists N: \forall n > N: \size {x_n} < H^{-1}$
That is:
:$\size {\dfrac 1 {x_n} } > H$.
So:
:$\exists N: \forall n > N: \size {\dfrac 1 {x_n} } > H$
and thus:
:$\sequence {\size {\dfrac 1 {x_n} } }$ diverges to ... | Let $\sequence {x_n}$ be a [[Definition:Real Sequence|sequence in $\R$]].
Let $\forall n \in \N: x_n > 0$.
Then:
:$x_n \to 0$ as $n \to \infty$ {{iff}} $\size {\dfrac 1 {x_n} } \to \infty$ as $n \to \infty$ | Suppose $x_n \to 0$ as $n \to \infty$.
Let $H > 0$.
So $H^{-1} > 0$.
Since $x_n \to 0$ as $n \to \infty$:
:$\exists N: \forall n > N: \size {x_n} < H^{-1}$
That is:
:$\size {\dfrac 1 {x_n} } > H$.
So:
:$\exists N: \forall n > N: \size {\dfrac 1 {x_n} } > H$
and thus:
:$\sequence {\size {\dfrac 1 {x_n} } }$ [[Defin... | Reciprocal of Null Sequence | https://proofwiki.org/wiki/Reciprocal_of_Null_Sequence | https://proofwiki.org/wiki/Reciprocal_of_Null_Sequence | [
"Limits of Sequences",
"Reciprocals",
"Reciprocal of Null Sequence"
] | [
"Definition:Real Sequence"
] | [
"Definition:Unbounded Divergent Sequence/Real Sequence/Positive Infinity"
] |
proofwiki-1468 | Modulus of Limit | Let $X$ be one of the standard number fields $\Q, \R, \C$.
Let $\sequence {x_n}$ be a sequence in $X$.
Let $\sequence {x_n}$ be convergent to the limit $l$.
That is, let $\ds \lim_{n \mathop \to \infty} x_n = l$.
Then
:$\ds \lim_{n \mathop \to \infty} \cmod {x_n} = \cmod l$
where $\cmod {x_n}$ is the modulus of $x_n$. | By the Triangle Inequality, we have:
:$\cmod {\cmod {x_n} - \cmod l} \le \cmod {x_n - l}$
Hence by the Squeeze Theorem and Convergent Sequence Minus Limit, $\cmod {x_n} \to \cmod l$ as $n \to \infty$.
{{Qed}} | Let $X$ be one of the [[Definition:Standard Number Field|standard number fields]] $\Q, \R, \C$.
Let $\sequence {x_n}$ be a [[Definition:Sequence|sequence in $X$]].
Let $\sequence {x_n}$ be [[Definition:Convergent Sequence (Analysis)|convergent]] to the [[Definition:Limit of Sequence (Number Field)|limit]] $l$.
That ... | By the [[Triangle Inequality]], we have:
:$\cmod {\cmod {x_n} - \cmod l} \le \cmod {x_n - l}$
Hence by the [[Squeeze Theorem]] and [[Convergent Sequence Minus Limit]], $\cmod {x_n} \to \cmod l$ as $n \to \infty$.
{{Qed}} | Modulus of Limit | https://proofwiki.org/wiki/Modulus_of_Limit | https://proofwiki.org/wiki/Modulus_of_Limit | [
"Modulus of Limit",
"Limits of Sequences",
"Modulus of Limit"
] | [
"Definition:Standard Number Field",
"Definition:Sequence",
"Definition:Convergent Sequence/Analysis",
"Definition:Limit of Sequence (Number Field)",
"Definition:Complex Modulus"
] | [
"Triangle Inequality",
"Squeeze Theorem",
"Convergent Sequence Minus Limit"
] |
proofwiki-1469 | Strictly Increasing Sequence of Natural Numbers | Let $\N_{>0}$ be the set of natural numbers without zero:
:$\N_{>0} = \set {1, 2, 3, \ldots}$
Let $\sequence {n_r}$ be a strictly increasing sequence in $\N_{>0}$.
Then:
:$\forall r \in \N_{>0}: n_r \ge r$ | This is to be proved by induction on $r$.
For all $r \in \N_{>0}$, let $\map P r$ be the proposition $n_r \ge r$. | Let $\N_{>0}$ be the set of [[Definition:Natural Numbers|natural numbers]] without zero:
:$\N_{>0} = \set {1, 2, 3, \ldots}$
Let $\sequence {n_r}$ be a [[Definition:Strictly Increasing Sequence|strictly increasing sequence]] in $\N_{>0}$.
Then:
:$\forall r \in \N_{>0}: n_r \ge r$ | This is to be proved by [[Principle of Mathematical Induction|induction]] on $r$.
For all $r \in \N_{>0}$, let $\map P r$ be the [[Definition:Proposition|proposition]] $n_r \ge r$. | Strictly Increasing Sequence of Natural Numbers | https://proofwiki.org/wiki/Strictly_Increasing_Sequence_of_Natural_Numbers | https://proofwiki.org/wiki/Strictly_Increasing_Sequence_of_Natural_Numbers | [
"Increasing Sequences",
"Proofs by Induction"
] | [
"Definition:Natural Numbers",
"Definition:Strictly Increasing/Sequence"
] | [
"Principle of Mathematical Induction",
"Definition:Proposition",
"Principle of Mathematical Induction"
] |
proofwiki-1470 | Limit of Subsequence equals Limit of Sequence | Let $T = \struct {S, \tau}$ be a topological space.
Let $\sequence {x_n}$ be a sequence in $T$.
Let $l \in S$ be a limit of $\sequence {x_n}$.
Let $\sequence {x_{n_r} }$ be a subsequence of $\sequence {x_n}$.
Then $l$ is a limit of $\sequence {x_{n_r} }$
That is, the limit of a convergent sequence in a topological spac... | Let $U \in \tau$ be an open set such that $l \in U$.
By definition of convergence, we have:
:$\exists N \in \N: \forall n > N: x_n \in U$.
When $r > N$, we have $n_r > n_N > N$ by Strictly Increasing Sequence of Natural Numbers.
It follows that:
:$\exists N \in \N: \forall r > N: x_{n_r} \in U$.
Therefore, as $U$ was a... | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $\sequence {x_n}$ be a [[Definition:Sequence|sequence in $T$]].
Let $l \in S$ be a [[Definition:Limit of Sequence (Topology)|limit]] of $\sequence {x_n}$.
Let $\sequence {x_{n_r} }$ be a [[Definition:Subsequence|subsequence]] of... | Let $U \in \tau$ be an [[Definition:Open Set (Topology)|open set]] such that $l \in U$.
By [[Definition:Convergent Sequence (Topology)|definition]] of convergence, we have:
:$\exists N \in \N: \forall n > N: x_n \in U$.
When $r > N$, we have $n_r > n_N > N$ by [[Strictly Increasing Sequence of Natural Numbers]].
It ... | Limit of Subsequence equals Limit of Sequence | https://proofwiki.org/wiki/Limit_of_Subsequence_equals_Limit_of_Sequence | https://proofwiki.org/wiki/Limit_of_Subsequence_equals_Limit_of_Sequence | [
"Convergence",
"Limits of Sequences",
"Subsequences"
] | [
"Definition:Topological Space",
"Definition:Sequence",
"Definition:Limit of Sequence/Topological Space",
"Definition:Subsequence",
"Definition:Limit of Sequence/Topological Space",
"Definition:Limit of Sequence/Topological Space",
"Definition:Convergent Sequence/Topology",
"Definition:Topological Spac... | [
"Definition:Open Set/Topology",
"Definition:Convergent Sequence/Topology",
"Strictly Increasing Sequence of Natural Numbers",
"Definition:Limit of Sequence/Topological Space",
"Definition:Convergent Sequence/Topology",
"Category:Convergence",
"Category:Limits of Sequences",
"Category:Subsequences"
] |
proofwiki-1471 | Root of Number Greater than One | Let $x \in \R$ be a real number.
Let $n \in \N_{>0}$ be a natural number such that $n > 0$.
Then:
:$x \ge 1 \implies x^{1/n} \ge 1$
where $x^{1/n}$ is the $n$th root of $x$. | Let $y = x^{1/n}$.
From the definition of the $n$th root of $x$, it follows that $x = y^n$.
We will show by induction that $\forall n \in \N_{>0}: y^n \ge 1 \implies y \ge 1$.
For all $n \in \N_{>0}$, let $\map P n$ be the proposition:
:$y^n \ge 1 \implies y \ge 1$ | Let $x \in \R$ be a [[Definition:Real Number|real number]].
Let $n \in \N_{>0}$ be a [[Definition:Natural Numbers|natural number]] such that $n > 0$.
Then:
:$x \ge 1 \implies x^{1/n} \ge 1$
where $x^{1/n}$ is the [[Definition:Root of Number|$n$th root]] of $x$. | Let $y = x^{1/n}$.
From the definition of the [[Definition:Root of Number|$n$th root]] of $x$, it follows that $x = y^n$.
We will show by [[Principle of Mathematical Induction|induction]] that $\forall n \in \N_{>0}: y^n \ge 1 \implies y \ge 1$.
For all $n \in \N_{>0}$, let $\map P n$ be the [[Definition:Propositi... | Root of Number Greater than One | https://proofwiki.org/wiki/Root_of_Number_Greater_than_One | https://proofwiki.org/wiki/Root_of_Number_Greater_than_One | [
"Analysis",
"Inequalities"
] | [
"Definition:Real Number",
"Definition:Natural Numbers",
"Definition:Root of Number"
] | [
"Definition:Root of Number",
"Principle of Mathematical Induction",
"Definition:Proposition",
"Principle of Mathematical Induction"
] |
proofwiki-1472 | Limit of Root of Positive Real Number | Let $x \in \R: x > 0$ be a real number.
Let $\sequence {x_n}$ be the sequence in $\R$ defined as:
:$x_n = x^{1 / n}$
Then $x_n \to 1$ as $n \to \infty$. | Let us define $a_1 = a_2 = \cdots = a_{n-1} = 1$ and $a_n = x$.
Let $G_n$ be the geometric mean of $a_1, \ldots, a_n$.
Let $A_n$ be the arithmetic mean of $a_1, \ldots, a_n$.
From their definitions:
:$G_n = x^{1/n}$
and:
:$A_n = \dfrac {n - 1 + x} n = 1 + \dfrac{x - 1} n$
From Arithmetic Mean is Never Less than Geometr... | Let $x \in \R: x > 0$ be a [[Definition:Real Number|real number]].
Let $\sequence {x_n}$ be the [[Definition:Real Sequence|sequence in $\R$]] defined as:
:$x_n = x^{1 / n}$
Then $x_n \to 1$ as $n \to \infty$. | Let us define $a_1 = a_2 = \cdots = a_{n-1} = 1$ and $a_n = x$.
Let $G_n$ be the [[Definition:Geometric Mean|geometric mean]] of $a_1, \ldots, a_n$.
Let $A_n$ be the [[Definition:Arithmetic Mean|arithmetic mean]] of $a_1, \ldots, a_n$.
From their definitions:
:$G_n = x^{1/n}$
and:
:$A_n = \dfrac {n - 1 + x} n = 1 + ... | Limit of Root of Positive Real Number/Proof 1 | https://proofwiki.org/wiki/Limit_of_Root_of_Positive_Real_Number | https://proofwiki.org/wiki/Limit_of_Root_of_Positive_Real_Number/Proof_1 | [
"Limits of Sequences",
"Limit of Root of Positive Real Number"
] | [
"Definition:Real Number",
"Definition:Real Sequence"
] | [
"Definition:Geometric Mean",
"Definition:Arithmetic Mean",
"Cauchy's Mean Theorem",
"Root of Number Greater than One",
"Sequence of Powers of Reciprocals is Null Sequence",
"Combination Theorem for Sequences",
"Squeeze Theorem",
"Combination Theorem for Sequences",
"Combination Theorem for Sequences... |
proofwiki-1473 | Limit of Root of Positive Real Number | Let $x \in \R: x > 0$ be a real number.
Let $\sequence {x_n}$ be the sequence in $\R$ defined as:
:$x_n = x^{1 / n}$
Then $x_n \to 1$ as $n \to \infty$. | We consider the case where $x \ge 1$; when $0 < x < 1$ the proof can be completed as for proof 1.
From Root of Number Greater than One:
:$x^{1/n} \ge 1$
Hence $\sequence {x^{1/n} }$ is bounded below by $1$.
Now consider $x^{1/n} / x^{1 / \paren {n + 1} }$:
{{begin-eqn}}
{{eqn | l = \frac {x^{1/n} } {x^{\frac 1 {n + 1} ... | Let $x \in \R: x > 0$ be a [[Definition:Real Number|real number]].
Let $\sequence {x_n}$ be the [[Definition:Real Sequence|sequence in $\R$]] defined as:
:$x_n = x^{1 / n}$
Then $x_n \to 1$ as $n \to \infty$. | We consider the case where $x \ge 1$; when $0 < x < 1$ the proof can be completed as for [[Limit of Root of Positive Real Number/Proof 1|proof 1]].
From [[Root of Number Greater than One]]:
:$x^{1/n} \ge 1$
Hence $\sequence {x^{1/n} }$ is [[Definition:Bounded Below Real Sequence|bounded below]] by $1$.
Now consider... | Limit of Root of Positive Real Number/Proof 2 | https://proofwiki.org/wiki/Limit_of_Root_of_Positive_Real_Number | https://proofwiki.org/wiki/Limit_of_Root_of_Positive_Real_Number/Proof_2 | [
"Limits of Sequences",
"Limit of Root of Positive Real Number"
] | [
"Definition:Real Number",
"Definition:Real Sequence"
] | [
"Limit of Root of Positive Real Number/Proof 1",
"Root of Number Greater than One",
"Definition:Bounded Below Sequence/Real",
"Root of Number Greater than One",
"Definition:Strictly Decreasing/Sequence/Real Sequence",
"Monotone Convergence Theorem (Real Analysis)",
"Definition:Convergent Sequence/Real N... |
proofwiki-1474 | Hero's Method | Let $a \in \R$ be a real number such that $a > 0$.
Let $x_1 \in \R$ be a real number such that $x_1 > 0$.
Let $\sequence {x_n}$ be the sequence in $\R$ defined recursively by:
:$\forall n \in \N_{>0}: x_{n + 1} = \dfrac {x_n + \dfrac a {x_n} } 2$
Then $x_n \to \sqrt a$ as $n \to \infty$. | Consider $x_n - x_{n + 1}$.
{{begin-eqn}}
{{eqn | l = x_n - x_{n + 1}
| r = x_n - \frac {x_n + \dfrac a {x_n} } 2
| c =
}}
{{eqn | r = \frac 1 {2 x_n} \paren {x_n^2 - a}
| c =
}}
{{eqn | o = \ge
| r = 0
| c = for $n \ge 2$
| cc= Lemma 2
}}
{{end-eqn}}
So, providing we ignore the fi... | Let $a \in \R$ be a [[Definition:Real Number|real number]] such that $a > 0$.
Let $x_1 \in \R$ be a [[Definition:Real Number|real number]] such that $x_1 > 0$.
Let $\sequence {x_n}$ be the [[Definition:Real Sequence|sequence in $\R$]] defined recursively by:
:$\forall n \in \N_{>0}: x_{n + 1} = \dfrac {x_n + \dfrac ... | Consider $x_n - x_{n + 1}$.
{{begin-eqn}}
{{eqn | l = x_n - x_{n + 1}
| r = x_n - \frac {x_n + \dfrac a {x_n} } 2
| c =
}}
{{eqn | r = \frac 1 {2 x_n} \paren {x_n^2 - a}
| c =
}}
{{eqn | o = \ge
| r = 0
| c = for $n \ge 2$
| cc= [[Hero's Method/Lemma 2|Lemma 2]]
}}
{{end-eqn}}
So... | Hero's Method/Proof 1 | https://proofwiki.org/wiki/Hero's_Method | https://proofwiki.org/wiki/Hero's_Method/Proof_1 | [
"Real Analysis",
"Square Roots",
"Hero's Method"
] | [
"Definition:Real Number",
"Definition:Real Number",
"Definition:Real Sequence"
] | [
"Hero's Method/Lemma 2",
"Definition:Decreasing/Sequence/Real Sequence",
"Definition:Bounded Below Sequence/Real",
"Monotone Convergence Theorem (Real Analysis)",
"Limit of Subsequence equals Limit of Sequence/Real Numbers",
"Combination Theorem for Sequences",
"Convergent Real Sequence has Unique Limit... |
proofwiki-1475 | Hero's Method | Let $a \in \R$ be a real number such that $a > 0$.
Let $x_1 \in \R$ be a real number such that $x_1 > 0$.
Let $\sequence {x_n}$ be the sequence in $\R$ defined recursively by:
:$\forall n \in \N_{>0}: x_{n + 1} = \dfrac {x_n + \dfrac a {x_n} } 2$
Then $x_n \to \sqrt a$ as $n \to \infty$. | Let $a > 0$.
We make no statement about $x_1$.
We specify that:
:$x_{n + 1} = \dfrac {x_n + \dfrac a {x_n} } 2$
Now:
{{begin-eqn}}
{{eqn | l = x_{n + 1} - \sqrt a
| r = \frac {x_n + \dfrac a {x_n} } 2 - \sqrt a
| c =
}}
{{eqn | r = \frac 1 {2 x_n} \paren {x_n^2 - 2 x_n \sqrt a + a}
| c =
}}
{{eqn | ... | Let $a \in \R$ be a [[Definition:Real Number|real number]] such that $a > 0$.
Let $x_1 \in \R$ be a [[Definition:Real Number|real number]] such that $x_1 > 0$.
Let $\sequence {x_n}$ be the [[Definition:Real Sequence|sequence in $\R$]] defined recursively by:
:$\forall n \in \N_{>0}: x_{n + 1} = \dfrac {x_n + \dfrac ... | Let $a > 0$.
We make no statement about $x_1$.
We specify that:
:$x_{n + 1} = \dfrac {x_n + \dfrac a {x_n} } 2$
Now:
{{begin-eqn}}
{{eqn | l = x_{n + 1} - \sqrt a
| r = \frac {x_n + \dfrac a {x_n} } 2 - \sqrt a
| c =
}}
{{eqn | r = \frac 1 {2 x_n} \paren {x_n^2 - 2 x_n \sqrt a + a}
| c =
}}
{{eq... | Hero's Method/Proof 2 | https://proofwiki.org/wiki/Hero's_Method | https://proofwiki.org/wiki/Hero's_Method/Proof_2 | [
"Real Analysis",
"Square Roots",
"Hero's Method"
] | [
"Definition:Real Number",
"Definition:Real Number",
"Definition:Real Sequence"
] | [
"Hero's Method/Lemma 2",
"Sequence of Powers of Number less than One",
"Limit of Subsequence equals Limit of Sequence/Real Numbers"
] |
proofwiki-1476 | Limit of Integer to Reciprocal Power | Let $\sequence {x_n}$ be the real sequence defined as $x_n = n^{1/n}$, using exponentiation.
Then $\sequence {x_n}$ converges with a limit of $1$. | From Number to Reciprocal Power is Decreasing we have that the real sequence $\sequence {n^{1/n} }$ is decreasing for $n \ge 3$.
Now, as $n^{1 / n} > 0$ for all positive $n$, it follows that $\sequence {n^{1 / n} }$ is bounded below (by $0$, for a start).
Thus the subsequence of $\sequence {n^{1 / n} }$ consisting of a... | Let $\sequence {x_n}$ be the [[Definition:Real Sequence|real sequence]] defined as $x_n = n^{1/n}$, using [[Definition:Real Exponential Function|exponentiation]].
Then $\sequence {x_n}$ [[Definition:Convergent Sequence|converges]] with a [[Definition:Limit of Sequence (Number Field)|limit]] of $1$. | From [[Number to Reciprocal Power is Decreasing]] we have that the [[Definition:Real Sequence|real sequence]] $\sequence {n^{1/n} }$ is [[Definition:Decreasing Real Sequence|decreasing]] for $n \ge 3$.
Now, as $n^{1 / n} > 0$ for all [[Definition:Positive Integer|positive]] $n$, it follows that $\sequence {n^{1 / n} }... | Limit of Integer to Reciprocal Power/Proof 1 | https://proofwiki.org/wiki/Limit_of_Integer_to_Reciprocal_Power | https://proofwiki.org/wiki/Limit_of_Integer_to_Reciprocal_Power/Proof_1 | [
"Limits of Sequences",
"Reciprocals",
"Limit of Integer to Reciprocal Power"
] | [
"Definition:Real Sequence",
"Definition:Exponential Function/Real",
"Definition:Convergent Sequence",
"Definition:Limit of Sequence (Number Field)"
] | [
"Number to Reciprocal Power is Decreasing",
"Definition:Real Sequence",
"Definition:Decreasing/Sequence/Real Sequence",
"Definition:Positive/Integer",
"Definition:Bounded Below Sequence/Real",
"Definition:Subsequence",
"Definition:Term of Sequence",
"Definition:Convergent Sequence/Real Numbers",
"Mo... |
proofwiki-1477 | Limit of Integer to Reciprocal Power | Let $\sequence {x_n}$ be the real sequence defined as $x_n = n^{1/n}$, using exponentiation.
Then $\sequence {x_n}$ converges with a limit of $1$. | We have the definition of the power to a real number:
:$\ds n^{1/n} = \map \exp {\frac 1 n \ln n}$
From Powers Drown Logarithms, we have that:
:$\ds \lim_{n \mathop \to \infty} \frac 1 n \ln n = 0$
Hence:
:$\ds \lim_{n \mathop \to \infty} n^{1/n} = \exp 0 = 1$
and the result.
{{qed}} | Let $\sequence {x_n}$ be the [[Definition:Real Sequence|real sequence]] defined as $x_n = n^{1/n}$, using [[Definition:Real Exponential Function|exponentiation]].
Then $\sequence {x_n}$ [[Definition:Convergent Sequence|converges]] with a [[Definition:Limit of Sequence (Number Field)|limit]] of $1$. | We have the definition of the [[Definition:Power to Real Number|power to a real number]]:
:$\ds n^{1/n} = \map \exp {\frac 1 n \ln n}$
From [[Powers Drown Logarithms]], we have that:
:$\ds \lim_{n \mathop \to \infty} \frac 1 n \ln n = 0$
Hence:
:$\ds \lim_{n \mathop \to \infty} n^{1/n} = \exp 0 = 1$
and the result.
{... | Limit of Integer to Reciprocal Power/Proof 2 | https://proofwiki.org/wiki/Limit_of_Integer_to_Reciprocal_Power | https://proofwiki.org/wiki/Limit_of_Integer_to_Reciprocal_Power/Proof_2 | [
"Limits of Sequences",
"Reciprocals",
"Limit of Integer to Reciprocal Power"
] | [
"Definition:Real Sequence",
"Definition:Exponential Function/Real",
"Definition:Convergent Sequence",
"Definition:Limit of Sequence (Number Field)"
] | [
"Definition:Power (Algebra)/Real Number",
"Powers Drown Logarithms"
] |
proofwiki-1478 | Limit of Integer to Reciprocal Power | Let $\sequence {x_n}$ be the real sequence defined as $x_n = n^{1/n}$, using exponentiation.
Then $\sequence {x_n}$ converges with a limit of $1$. | Let $n^{1/n} = 1 + a_n$.
The strategy is to:
:$(1): \quad$ prove that $a_n > 0$ for $n > 1$
:$(2): \quad$ deduce that $n - 1 \ge \dfrac {n \paren {n - 1} } {2!} a_n^2$ for $n > 1$
and hence:
:$(3): \quad$ deduce that $0 \le a_n^2 \le \dfrac 2 n$
Let $n > 1$.
Then:
{{begin-eqn}}
{{eqn | l = n
| r = \paren {1 + a_n... | Let $\sequence {x_n}$ be the [[Definition:Real Sequence|real sequence]] defined as $x_n = n^{1/n}$, using [[Definition:Real Exponential Function|exponentiation]].
Then $\sequence {x_n}$ [[Definition:Convergent Sequence|converges]] with a [[Definition:Limit of Sequence (Number Field)|limit]] of $1$. | Let $n^{1/n} = 1 + a_n$.
The strategy is to:
:$(1): \quad$ prove that $a_n > 0$ for $n > 1$
:$(2): \quad$ deduce that $n - 1 \ge \dfrac {n \paren {n - 1} } {2!} a_n^2$ for $n > 1$
and hence:
:$(3): \quad$ deduce that $0 \le a_n^2 \le \dfrac 2 n$
Let $n > 1$.
Then:
{{begin-eqn}}
{{eqn | l = n
| r = \paren {1 ... | Limit of Integer to Reciprocal Power/Proof 3 | https://proofwiki.org/wiki/Limit_of_Integer_to_Reciprocal_Power | https://proofwiki.org/wiki/Limit_of_Integer_to_Reciprocal_Power/Proof_3 | [
"Limits of Sequences",
"Reciprocals",
"Limit of Integer to Reciprocal Power"
] | [
"Definition:Real Sequence",
"Definition:Exponential Function/Real",
"Definition:Convergent Sequence",
"Definition:Limit of Sequence (Number Field)"
] | [] |
proofwiki-1479 | Difference Between Adjacent Square Roots Converges | Let $\sequence {x_n}$ be the sequence in $\R$ defined as $x_n = \sqrt {n + 1} - \sqrt n$.
Then $\sequence {x_n}$ converges to a zero limit. | We have:
{{begin-eqn}}
{{eqn | l = 0
| o = \le
| r = \sqrt {n + 1} - \sqrt n
| c =
}}
{{eqn | r = \frac {\paren {\sqrt {n + 1} - \sqrt n} \paren {\sqrt {n + 1} + \sqrt n} } {\sqrt {n + 1} + \sqrt n}
| c = multiplying top and bottom by $\sqrt {n + 1} + \sqrt n$
}}
{{eqn | r = \frac {n + 1 - n} {... | Let $\sequence {x_n}$ be the [[Definition:Real Sequence|sequence in $\R$]] defined as $x_n = \sqrt {n + 1} - \sqrt n$.
Then $\sequence {x_n}$ [[Definition:Convergent Sequence|converges]] to a zero [[Definition:Limit of a Sequence (Number Field)|limit]]. | We have:
{{begin-eqn}}
{{eqn | l = 0
| o = \le
| r = \sqrt {n + 1} - \sqrt n
| c =
}}
{{eqn | r = \frac {\paren {\sqrt {n + 1} - \sqrt n} \paren {\sqrt {n + 1} + \sqrt n} } {\sqrt {n + 1} + \sqrt n}
| c = multiplying top and bottom by $\sqrt {n + 1} + \sqrt n$
}}
{{eqn | r = \frac {n + 1 - n} ... | Difference Between Adjacent Square Roots Converges | https://proofwiki.org/wiki/Difference_Between_Adjacent_Square_Roots_Converges | https://proofwiki.org/wiki/Difference_Between_Adjacent_Square_Roots_Converges | [
"Limits of Sequences"
] | [
"Definition:Real Sequence",
"Definition:Convergent Sequence",
"Definition:Limit of Sequence (Number Field)"
] | [
"Difference of Two Squares",
"Sequence of Powers of Reciprocals is Null Sequence",
"Squeeze Theorem"
] |
proofwiki-1480 | Peak Point Lemma | Let $\sequence {x_n}$ be a sequence in $\R$ which is infinite.
Then $\sequence {x_n}$ has an infinite subsequence which is monotone. | There are $2$ cases to consider.
First, suppose that every set $\set {x_n: n > N}$ has a maximum.
If this is the case, we can find a sequence $n_r \in \N$ such that:
:$\ds x_{n_1} = \max \set {x_n: n > 1}$
:$\ds x_{n_2} = \max \set {x_n: n > n_1}$
:$\ds x_{n_3} = \max \set {x_n: n > n_2}$
and so on.
From the method of ... | Let $\sequence {x_n}$ be a [[Definition:Real Sequence|sequence in $\R$]] which is [[Definition:Infinite Sequence|infinite]].
Then $\sequence {x_n}$ has an [[Definition:Infinite Sequence|infinite]] [[Definition:Subsequence|subsequence]] which is [[Definition:Monotone Sequence|monotone]]. | There are $2$ cases to consider.
First, suppose that every [[Definition:Set|set]] $\set {x_n: n > N}$ has a [[Definition:Greatest Element|maximum]].
If this is the case, we can find a [[Definition:Real Sequence|sequence]] $n_r \in \N$ such that:
:$\ds x_{n_1} = \max \set {x_n: n > 1}$
:$\ds x_{n_2} = \max \set {x_n... | Peak Point Lemma/Proof 1 | https://proofwiki.org/wiki/Peak_Point_Lemma | https://proofwiki.org/wiki/Peak_Point_Lemma/Proof_1 | [
"Limits of Sequences",
"Named Theorems",
"Peak Point Lemma"
] | [
"Definition:Real Sequence",
"Definition:Sequence/Infinite Sequence",
"Definition:Sequence/Infinite Sequence",
"Definition:Subsequence",
"Definition:Monotone (Order Theory)/Sequence"
] | [
"Definition:Set",
"Definition:Greatest Element",
"Definition:Real Sequence",
"Definition:Greatest Element",
"Definition:Subset",
"Definition:Set",
"Definition:Greatest Element",
"Definition:Term of Sequence",
"Definition:Real Sequence",
"Definition:Decreasing/Sequence/Real Sequence",
"Definition... |
proofwiki-1481 | Peak Point Lemma | Let $\sequence {x_n}$ be a sequence in $\R$ which is infinite.
Then $\sequence {x_n}$ has an infinite subsequence which is monotone. | A coastal town in Spain has an infinite row of hotels along a road leading down to the beach.
The tourist bureau which rates these hotels has a special designation for any hotel having a view of the sea.
Any hotel which is at least as tall as the rest of the hotels on the road to the sea receives this view-rating.
Star... | Let $\sequence {x_n}$ be a [[Definition:Real Sequence|sequence in $\R$]] which is [[Definition:Infinite Sequence|infinite]].
Then $\sequence {x_n}$ has an [[Definition:Infinite Sequence|infinite]] [[Definition:Subsequence|subsequence]] which is [[Definition:Monotone Sequence|monotone]]. | A coastal town in Spain has an infinite row of hotels along a road leading down to the beach.
The tourist bureau which rates these hotels has a special designation for any hotel having a view of the sea.
Any hotel which is at least as tall as the rest of the hotels on the road to the sea receives this view-rating.
S... | Peak Point Lemma/Proof 2 | https://proofwiki.org/wiki/Peak_Point_Lemma | https://proofwiki.org/wiki/Peak_Point_Lemma/Proof_2 | [
"Limits of Sequences",
"Named Theorems",
"Peak Point Lemma"
] | [
"Definition:Real Sequence",
"Definition:Sequence/Infinite Sequence",
"Definition:Sequence/Infinite Sequence",
"Definition:Subsequence",
"Definition:Monotone (Order Theory)/Sequence"
] | [
"Definition:Decreasing/Sequence",
"Definition:Subsequence",
"Definition:Increasing/Sequence",
"Definition:Subsequence",
"Definition:Increasing/Sequence",
"Definition:Subsequence"
] |
proofwiki-1482 | Bolzano-Weierstrass Theorem | Every bounded sequence of real numbers has a convergent subsequence. | Let $\sequence {x_n}$ be a bounded sequence in $\R$.
By the Peak Point Lemma, $\sequence {x_n}$ has a monotone subsequence $\sequence {x_{n_r} }$.
Since $\sequence {x_n}$ is bounded, so is $\sequence {x_{n_r} }$.
Hence, by the Monotone Convergence Theorem (Real Analysis), the result follows.
{{qed}} | Every [[Definition:Bounded Real Sequence|bounded sequence]] of [[Definition:Real Number|real numbers]] has a [[Definition:Convergent Real Sequence|convergent]] [[Definition:Subsequence|subsequence]]. | Let $\sequence {x_n}$ be a [[Definition:Bounded Real Sequence|bounded sequence in $\R$]].
By the [[Peak Point Lemma]], $\sequence {x_n}$ has a [[Definition:Monotone Real Sequence|monotone]] [[Definition:Subsequence|subsequence]] $\sequence {x_{n_r} }$.
Since $\sequence {x_n}$ is [[Definition:Bounded Sequence|bounded]... | Bolzano-Weierstrass Theorem/Proof 1 | https://proofwiki.org/wiki/Bolzano-Weierstrass_Theorem | https://proofwiki.org/wiki/Bolzano-Weierstrass_Theorem/Proof_1 | [
"Bolzano-Weierstrass Theorem",
"Limits of Sequences",
"Real Analysis"
] | [
"Definition:Bounded Sequence/Real",
"Definition:Real Number",
"Definition:Convergent Sequence/Real Numbers",
"Definition:Subsequence"
] | [
"Definition:Bounded Sequence/Real",
"Peak Point Lemma",
"Definition:Monotone (Order Theory)/Sequence/Real Sequence",
"Definition:Subsequence",
"Definition:Bounded Sequence",
"Monotone Convergence Theorem (Real Analysis)"
] |
proofwiki-1483 | Bolzano-Weierstrass Theorem | Every bounded sequence of real numbers has a convergent subsequence. | Let $\sequence {x_n}_{n \mathop \in \N}$ be a bounded sequence in $\R$.
By definition there are real numbers $a, b \in \R$ such that $ x_n \in \openint{a}{b}$ for all $n \in \N$.
We will construct a subsequence $\sequence {x_{n_i}}_{i \mathop \in \N}$ and two sequences of real numbers $\sequence {b_i}_{i \mathop \in \... | Every [[Definition:Bounded Real Sequence|bounded sequence]] of [[Definition:Real Number|real numbers]] has a [[Definition:Convergent Real Sequence|convergent]] [[Definition:Subsequence|subsequence]]. | Let $\sequence {x_n}_{n \mathop \in \N}$ be a [[Definition:Bounded Real Sequence|bounded]] [[Definition:Real Sequence|sequence in $\R$]].
By definition there are [[Definition:Real Number|real numbers]] $a, b \in \R$ such that $ x_n \in \openint{a}{b}$ for all $n \in \N$.
We will construct a [[Definition:Subsequence|s... | Bolzano-Weierstrass Theorem/Proof 2 | https://proofwiki.org/wiki/Bolzano-Weierstrass_Theorem | https://proofwiki.org/wiki/Bolzano-Weierstrass_Theorem/Proof_2 | [
"Bolzano-Weierstrass Theorem",
"Limits of Sequences",
"Real Analysis"
] | [
"Definition:Bounded Sequence/Real",
"Definition:Real Number",
"Definition:Convergent Sequence/Real Numbers",
"Definition:Subsequence"
] | [
"Definition:Bounded Sequence/Real",
"Definition:Real Sequence",
"Definition:Real Number",
"Definition:Subsequence",
"Definition:Real Number",
"Definition:Infinite Set",
"Definition:Set",
"Definition:Infinite Set",
"Definition:Convergent Sequence/Real Numbers",
"Definition:Indexing Set/Indexed Set"... |
proofwiki-1484 | Existence of Maximum and Minimum of Bounded Sequence | Let $\sequence {x_n}$ be a bounded sequence in $\R$ (which may or may not be convergent).
Let $L$ be the set of all real numbers which are the limit of some subsequence of $\sequence {x_n}$.
Then $L$ has both a maximum and a minimum. | From the Bolzano-Weierstrass Theorem:
:$L \ne \O$
From Lower and Upper Bounds for Sequences, $L$ is a bounded subset of $\R$.
Thus $L$ does have a supremum and infimum in $\R$.
The object of this proof is to confirm that:
:$\overline l := \map \sup L \in L$
and:
:$\underline l := \map \inf L \in L$
that is, that these ... | Let $\sequence {x_n}$ be a [[Definition:Bounded Real Sequence|bounded sequence in $\R$]] (which may or may not be [[Definition:Convergent Real Sequence|convergent]]).
Let $L$ be the [[Definition:Set|set]] of all [[Definition:Real Number|real numbers]] which are the [[Definition:Limit of Real Sequence|limit]] of some [... | From the [[Bolzano-Weierstrass Theorem]]:
:$L \ne \O$
From [[Lower and Upper Bounds for Sequences]], $L$ is a [[Definition:Bounded Subset of Real Numbers|bounded subset of $\R$]].
Thus $L$ does have a [[Definition:Supremum of Set|supremum]] and [[Definition:Infimum of Set|infimum]] in $\R$.
The object of this proof ... | Existence of Maximum and Minimum of Bounded Sequence | https://proofwiki.org/wiki/Existence_of_Maximum_and_Minimum_of_Bounded_Sequence | https://proofwiki.org/wiki/Existence_of_Maximum_and_Minimum_of_Bounded_Sequence | [
"Limits of Sequences"
] | [
"Definition:Bounded Sequence/Real",
"Definition:Convergent Sequence/Real Numbers",
"Definition:Set",
"Definition:Real Number",
"Definition:Limit of Sequence/Real Numbers",
"Definition:Subsequence",
"Definition:Greatest Element",
"Definition:Smallest Element"
] | [
"Bolzano-Weierstrass Theorem",
"Lower and Upper Bounds for Sequences",
"Definition:Bounded Set/Real Numbers",
"Definition:Supremum of Set",
"Definition:Infimum of Set",
"Definition:Subsequence",
"Definition:Supremum of Set",
"Definition:Upper Bound of Set",
"Definition:Upper Bound of Set",
"Defini... |
proofwiki-1485 | Difference of Two Powers | Let $\mathbb F$ denote one of the standard number systems, that is $\Z$, $\Q$, $\R$ and $\C$.
Let $n \in \N$ such that $n \ge 2$.
Then for all $a, b \in \mathbb F$:
{{begin-eqn}}
{{eqn | l = a^n - b^n
| r = \paren {a - b} \sum_{j \mathop = 0}^{n - 1} a^{n - j - 1} b^j
| c =
}}
{{eqn | r = \paren {a - b} \p... | Let $\ds S_n = \sum_{j \mathop = 0}^{n - 1} a^{n - j - 1} b^j$.
This can also be written:
:$\ds S_n = \sum_{j \mathop = 0}^{n - 1} b^j a^{n - j - 1}$
Consider:
:$\ds a S_n = \sum_{j \mathop = 0}^{n - 1} a^{n - j} b^j$
Taking the first term (where $j = 0$) out of the summation, we get:
:$\ds a S_n = \sum_{j \mathop = 0... | Let $\mathbb F$ denote one of the [[Definition:Standard Number System|standard number systems]], that is $\Z$, $\Q$, $\R$ and $\C$.
Let $n \in \N$ such that $n \ge 2$.
Then for all $a, b \in \mathbb F$:
{{begin-eqn}}
{{eqn | l = a^n - b^n
| r = \paren {a - b} \sum_{j \mathop = 0}^{n - 1} a^{n - j - 1} b^j
... | Let $\ds S_n = \sum_{j \mathop = 0}^{n - 1} a^{n - j - 1} b^j$.
This can also be written:
:$\ds S_n = \sum_{j \mathop = 0}^{n - 1} b^j a^{n - j - 1}$
Consider:
:$\ds a S_n = \sum_{j \mathop = 0}^{n - 1} a^{n - j} b^j$
Taking the first term (where $j = 0$) out of the summation, we get:
:$\ds a S_n = \sum_{j \mathop... | Difference of Two Powers/Proof 1 | https://proofwiki.org/wiki/Difference_of_Two_Powers | https://proofwiki.org/wiki/Difference_of_Two_Powers/Proof_1 | [
"Algebra",
"Polynomial Theory",
"Difference of Two Powers"
] | [
"Definition:Number"
] | [
"Permutation of Indices of Summation"
] |
proofwiki-1486 | Difference of Two Powers | Let $\mathbb F$ denote one of the standard number systems, that is $\Z$, $\Q$, $\R$ and $\C$.
Let $n \in \N$ such that $n \ge 2$.
Then for all $a, b \in \mathbb F$:
{{begin-eqn}}
{{eqn | l = a^n - b^n
| r = \paren {a - b} \sum_{j \mathop = 0}^{n - 1} a^{n - j - 1} b^j
| c =
}}
{{eqn | r = \paren {a - b} \p... | From Sum of Geometric Sequence:
{{begin-eqn}}
{{eqn | l = \sum_{j \mathop = 0}^{n - 1} x^j
| r = \frac {x^n - 1} {x - 1}
| c =
}}
{{eqn | ll= \leadsto
| l = \paren {\dfrac a b}^n - 1
| r = \paren {\dfrac a b - 1} \sum_{j \mathop = 0}^{n - 1} \paren {\dfrac a b}^j
| c = setting $x = \dfrac... | Let $\mathbb F$ denote one of the [[Definition:Standard Number System|standard number systems]], that is $\Z$, $\Q$, $\R$ and $\C$.
Let $n \in \N$ such that $n \ge 2$.
Then for all $a, b \in \mathbb F$:
{{begin-eqn}}
{{eqn | l = a^n - b^n
| r = \paren {a - b} \sum_{j \mathop = 0}^{n - 1} a^{n - j - 1} b^j
... | From [[Sum of Geometric Sequence]]:
{{begin-eqn}}
{{eqn | l = \sum_{j \mathop = 0}^{n - 1} x^j
| r = \frac {x^n - 1} {x - 1}
| c =
}}
{{eqn | ll= \leadsto
| l = \paren {\dfrac a b}^n - 1
| r = \paren {\dfrac a b - 1} \sum_{j \mathop = 0}^{n - 1} \paren {\dfrac a b}^j
| c = setting $x = ... | Difference of Two Powers/Proof 3 | https://proofwiki.org/wiki/Difference_of_Two_Powers | https://proofwiki.org/wiki/Difference_of_Two_Powers/Proof_3 | [
"Algebra",
"Polynomial Theory",
"Difference of Two Powers"
] | [
"Definition:Number"
] | [
"Sum of Geometric Sequence"
] |
proofwiki-1487 | Difference of Two Powers | Let $\mathbb F$ denote one of the standard number systems, that is $\Z$, $\Q$, $\R$ and $\C$.
Let $n \in \N$ such that $n \ge 2$.
Then for all $a, b \in \mathbb F$:
{{begin-eqn}}
{{eqn | l = a^n - b^n
| r = \paren {a - b} \sum_{j \mathop = 0}^{n - 1} a^{n - j - 1} b^j
| c =
}}
{{eqn | r = \paren {a - b} \p... | The proof will proceed by the Principle of Complete Finite Induction on $\Z_{>0}$.
Let $S$ be the set defined as:
:$\ds S := \set {n \in \Z_{>0}: a^n - b^n = \paren {a - b} \sum_{j \mathop = 0}^{n - 1} a^{n - j - 1} b^j}$
That is, $S$ is to be the set of all $n$ such that:
:$\ds a^n - b^n = \paren {a - b} \sum_{j \math... | Let $\mathbb F$ denote one of the [[Definition:Standard Number System|standard number systems]], that is $\Z$, $\Q$, $\R$ and $\C$.
Let $n \in \N$ such that $n \ge 2$.
Then for all $a, b \in \mathbb F$:
{{begin-eqn}}
{{eqn | l = a^n - b^n
| r = \paren {a - b} \sum_{j \mathop = 0}^{n - 1} a^{n - j - 1} b^j
... | The proof will proceed by the [[Principle of Complete Finite Induction]] on $\Z_{>0}$.
Let $S$ be the [[Definition:Set|set]] defined as:
:$\ds S := \set {n \in \Z_{>0}: a^n - b^n = \paren {a - b} \sum_{j \mathop = 0}^{n - 1} a^{n - j - 1} b^j}$
That is, $S$ is to be the [[Definition:Set|set]] of all $n$ such that:
:$... | Difference of Two Powers/Proof 4 | https://proofwiki.org/wiki/Difference_of_Two_Powers | https://proofwiki.org/wiki/Difference_of_Two_Powers/Proof_4 | [
"Algebra",
"Polynomial Theory",
"Difference of Two Powers"
] | [
"Definition:Number"
] | [
"Second Principle of Finite Induction",
"Definition:Set",
"Definition:Set",
"Definition:Basis for the Induction",
"Definition:Induction Hypothesis",
"Definition:Induction Step",
"Difference of Two Powers/Proof 4",
"Second Principle of Finite Induction"
] |
proofwiki-1488 | Difference of Two Powers | Let $\mathbb F$ denote one of the standard number systems, that is $\Z$, $\Q$, $\R$ and $\C$.
Let $n \in \N$ such that $n \ge 2$.
Then for all $a, b \in \mathbb F$:
{{begin-eqn}}
{{eqn | l = a^n - b^n
| r = \paren {a - b} \sum_{j \mathop = 0}^{n - 1} a^{n - j - 1} b^j
| c =
}}
{{eqn | r = \paren {a - b} \p... | Let $\map f x = x^n$ for $x \in \R, n \in \N$.
Let $a \in \R$.
By definition of the derivative:
:$\ds \map {f'} a = \lim_{x \mathop \to a} \frac {\map f x - \map f a} {x - a} = \lim_{x \mathop \to a} \frac {x^n - a^n} {x - a}$
=== Case $\text I$ ===
For $n = 0$ it is possible to do:
{{begin-eqn}}
{{eqn | l = \map {f'} ... | Let $\mathbb F$ denote one of the [[Definition:Standard Number System|standard number systems]], that is $\Z$, $\Q$, $\R$ and $\C$.
Let $n \in \N$ such that $n \ge 2$.
Then for all $a, b \in \mathbb F$:
{{begin-eqn}}
{{eqn | l = a^n - b^n
| r = \paren {a - b} \sum_{j \mathop = 0}^{n - 1} a^{n - j - 1} b^j
... | Let $\map f x = x^n$ for $x \in \R, n \in \N$.
Let $a \in \R$.
By definition of the [[Definition:Derivative|derivative]]:
:$\ds \map {f'} a = \lim_{x \mathop \to a} \frac {\map f x - \map f a} {x - a} = \lim_{x \mathop \to a} \frac {x^n - a^n} {x - a}$
=== Case $\text I$ ===
For $n = 0$ it is possible to do:
{{be... | Power Rule for Derivatives/Natural Number Index/Proof by Difference of Two Powers | https://proofwiki.org/wiki/Difference_of_Two_Powers | https://proofwiki.org/wiki/Power_Rule_for_Derivatives/Natural_Number_Index/Proof_by_Difference_of_Two_Powers | [
"Algebra",
"Polynomial Theory",
"Difference of Two Powers"
] | [
"Definition:Number"
] | [
"Definition:Derivative",
"Derivative of Identity Function/Real",
"Definition:Commutative Ring",
"Difference of Two Powers",
"Real Polynomial Function is Continuous"
] |
proofwiki-1489 | Polynomial Factor Theorem | Let $\map P x$ be a polynomial in $x$ over a field $K$ of degree $n$.
Then:
:$\xi \in K: \map P \xi = 0 \iff \map P x = \paren {x - \xi} \map Q x$
where $Q$ is a polynomial of degree $n - 1$.
Hence, if $\xi_1, \xi_2, \ldots, \xi_n \in K$ such that all are different, and $\map P {\xi_1} = \map P {\xi_2} = \dotsb = \map ... | Let $P = \paren {x - \xi} Q$.
Then:
:$\map P \xi = \map Q \xi \cdot 0 = 0$
Conversely, let $\map P \xi = 0$.
By the Division Theorem for Polynomial Forms over Field, there exist polynomials $Q$ and $R$ such that:
:$P = \paren {x - \xi} Q + R$
and:
:$\map \deg R < \map \deg {x - \xi} = 1$
Evaluating at $\xi$ we have:
:$... | Let $\map P x$ be a [[Definition:Polynomial over Field|polynomial]] in $x$ over a [[Definition:Field (Abstract Algebra)|field]] $K$ of [[Definition:Degree of Polynomial|degree]] $n$.
Then:
:$\xi \in K: \map P \xi = 0 \iff \map P x = \paren {x - \xi} \map Q x$
where $Q$ is a [[Definition:Polynomial over Field|polynomia... | Let $P = \paren {x - \xi} Q$.
Then:
:$\map P \xi = \map Q \xi \cdot 0 = 0$
Conversely, let $\map P \xi = 0$.
By the [[Division Theorem for Polynomial Forms over Field]], there exist polynomials $Q$ and $R$ such that:
:$P = \paren {x - \xi} Q + R$
and:
:$\map \deg R < \map \deg {x - \xi} = 1$
Evaluating at $\xi$ we... | Polynomial Factor Theorem | https://proofwiki.org/wiki/Polynomial_Factor_Theorem | https://proofwiki.org/wiki/Polynomial_Factor_Theorem | [
"Polynomial Factor Theorem",
"Polynomial Theory",
"Named Theorems"
] | [
"Definition:Polynomial over Ring",
"Definition:Field (Abstract Algebra)",
"Definition:Degree of Polynomial",
"Definition:Polynomial over Ring",
"Definition:Degree of Polynomial"
] | [
"Division Theorem for Polynomial Forms over Field",
"Ring of Polynomial Forms is Integral Domain",
"Degree of Product of Polynomials over Ring/Corollary 2",
"Definition:Degree of Polynomial/Zero",
"Definition:Constant Polynomial"
] |
proofwiki-1490 | Telescoping Series/Example 1 | Let $\sequence {b_n}$ be a sequence in $\R$.
Let $\sequence {a_n}$ be a sequence whose terms are defined as:
:$a_k = b_k - b_{k + 1}$
Then:
:$\ds \sum_{k \mathop = 1}^n a_k = b_1 - b_{n + 1}$ | {{begin-eqn}}
{{eqn | l = \ds \sum_{k \mathop = 1}^n a_k
| r = \sum_{k \mathop = 1}^n \paren {b_k - b_{k + 1} }
| c =
}}
{{eqn | r = \sum_{k \mathop = 1}^n b_k - \sum_{k \mathop = 1}^n b_{k + 1}
| c =
}}
{{eqn | r = \sum_{k \mathop = 1}^n b_k - \sum_{k \mathop = 2}^{n + 1} b_k
| c = Translatio... | Let $\sequence {b_n}$ be a [[Definition:Real Sequence|sequence in $\R$]].
Let $\sequence {a_n}$ be a [[Definition:Real Sequence|sequence]] whose [[Definition:Term of Sequence|terms]] are defined as:
:$a_k = b_k - b_{k + 1}$
Then:
:$\ds \sum_{k \mathop = 1}^n a_k = b_1 - b_{n + 1}$ | {{begin-eqn}}
{{eqn | l = \ds \sum_{k \mathop = 1}^n a_k
| r = \sum_{k \mathop = 1}^n \paren {b_k - b_{k + 1} }
| c =
}}
{{eqn | r = \sum_{k \mathop = 1}^n b_k - \sum_{k \mathop = 1}^n b_{k + 1}
| c =
}}
{{eqn | r = \sum_{k \mathop = 1}^n b_k - \sum_{k \mathop = 2}^{n + 1} b_k
| c = [[Translat... | Telescoping Series/Example 1 | https://proofwiki.org/wiki/Telescoping_Series/Example_1 | https://proofwiki.org/wiki/Telescoping_Series/Example_1 | [
"Telescoping Series"
] | [
"Definition:Real Sequence",
"Definition:Real Sequence",
"Definition:Term of Sequence"
] | [
"Translation of Index Variable of Summation",
"Definition:Convergent Sequence"
] |
proofwiki-1491 | Terms in Convergent Series Converge to Zero | Let $\sequence {a_n}$ be a sequence in any of the standard number fields $\Q$, $\R$, or $\C$.
Suppose that the series $\ds \sum_{n \mathop = 1}^\infty a_n$ converges in any of the standard number fields $\Q$, $\R$, or $\C$.
Then:
:$\ds \lim_{n \mathop \to \infty} a_n = 0$ | Let $\ds s = \sum_{n \mathop = 1}^\infty a_n$.
Then $\ds s_N = \sum_{n \mathop = 1}^N a_n \to s$ as $N \to \infty$.
Also, $s_{N - 1} \to s$ as $N \to \infty$.
Thus:
{{begin-eqn}}
{{eqn | l = a_N
| r = \paren {a_1 + a_2 + \cdots + a_{N - 1} + a_N} - \paren {a_1 + a_2 + \cdots + a_{N - 1} }
| c =
}}
{{eqn | ... | Let $\sequence {a_n}$ be a [[Definition:Sequence|sequence]] in any of the [[Definition:Standard Number Field|standard number fields]] [[Definition:Rational Number|$\Q$]], [[Definition:Real Number|$\R$]], or [[Definition:Complex Number|$\C$]].
Suppose that the [[Definition:Series|series]] $\ds \sum_{n \mathop = 1}^\inf... | Let $\ds s = \sum_{n \mathop = 1}^\infty a_n$.
Then $\ds s_N = \sum_{n \mathop = 1}^N a_n \to s$ as $N \to \infty$.
Also, $s_{N - 1} \to s$ as $N \to \infty$.
Thus:
{{begin-eqn}}
{{eqn | l = a_N
| r = \paren {a_1 + a_2 + \cdots + a_{N - 1} + a_N} - \paren {a_1 + a_2 + \cdots + a_{N - 1} }
| c =
}}
{{e... | Terms in Convergent Series Converge to Zero | https://proofwiki.org/wiki/Terms_in_Convergent_Series_Converge_to_Zero | https://proofwiki.org/wiki/Terms_in_Convergent_Series_Converge_to_Zero | [
"Convergent Series",
"Series"
] | [
"Definition:Sequence",
"Definition:Standard Number Field",
"Definition:Rational Number",
"Definition:Real Number",
"Definition:Complex Number",
"Definition:Series",
"Definition:Convergent Series",
"Definition:Standard Number Field",
"Definition:Rational Number",
"Definition:Real Number",
"Defini... | [] |
proofwiki-1492 | Linear Combination of Convergent Series | Let $\sequence {a_n}_{n \mathop \ge 1}$ and $\sequence {b_n}_{n \mathop \ge 1}$ be sequences of real numbers.
Let the two series $\ds\sum_{n \mathop = 1}^\infty a_n$ and $\ds \sum_{n \mathop = 1}^\infty b_n$ converge to $\alpha$ and $\beta$ respectively.
Let $\lambda, \mu \in \R$ be real numbers.
Then the series $\ds \... | {{begin-eqn}}
{{eqn | l = \sum_{n \mathop = 1}^N \paren {\lambda a_n + \mu b_n}
| r = \lambda \sum_{n \mathop = 1}^N a_n + \mu \sum_{n \mathop = 1}^N b_n
| c = Linear Combination of Indexed Summations
}}
{{eqn | o = \to
| r = \lambda \alpha + \mu \beta \text{ as } N \to \infty
| c = Combination ... | Let $\sequence {a_n}_{n \mathop \ge 1}$ and $\sequence {b_n}_{n \mathop \ge 1}$ be [[Definition:Sequence|sequences]] of [[Definition:Real Number|real numbers]].
Let the two [[Definition:Series|series]] $\ds\sum_{n \mathop = 1}^\infty a_n$ and $\ds \sum_{n \mathop = 1}^\infty b_n$ [[Definition:Convergent Series|converg... | {{begin-eqn}}
{{eqn | l = \sum_{n \mathop = 1}^N \paren {\lambda a_n + \mu b_n}
| r = \lambda \sum_{n \mathop = 1}^N a_n + \mu \sum_{n \mathop = 1}^N b_n
| c = [[Linear Combination of Indexed Summations]]
}}
{{eqn | o = \to
| r = \lambda \alpha + \mu \beta \text{ as } N \to \infty
| c = [[Combin... | Linear Combination of Convergent Series | https://proofwiki.org/wiki/Linear_Combination_of_Convergent_Series | https://proofwiki.org/wiki/Linear_Combination_of_Convergent_Series | [
"Series"
] | [
"Definition:Sequence",
"Definition:Real Number",
"Definition:Series",
"Definition:Convergent Series",
"Definition:Real Number",
"Definition:Series",
"Definition:Convergent Series"
] | [
"Linear Combination of Indexed Summations",
"Combination Theorem for Sequences"
] |
proofwiki-1493 | Abel's Limit Theorem | Let $\ds \sum_{k \mathop = 0}^\infty a_k$ be a convergent series in $\R$.
Then:
:$\ds \lim_{x \mathop \to 1^-} \paren {\sum_{k \mathop = 0}^\infty a_k x^k} = \sum_{k \mathop = 0}^\infty a_k$
where $\ds \lim_{x \mathop \to 1^-}$ denotes the limit from the left. | Let $\epsilon > 0$.
Let $\ds \sum_{k \mathop = 0}^\infty a_k$ converge to $s$.
Then its sequence of partial sums $\sequence {s_N}$, where $\ds s_N = \sum_{n \mathop = 1}^N a_n$, is a Cauchy sequence.
So:
:$\ds \exists N: \forall k, m: k \ge m \ge N: \size {\sum_{l \mathop = m}^k a_l} < \frac \epsilon 3$
From Abel's Lem... | Let $\ds \sum_{k \mathop = 0}^\infty a_k$ be a [[Definition:Convergent Series|convergent]] [[Definition:Series|series in $\R$]].
Then:
:$\ds \lim_{x \mathop \to 1^-} \paren {\sum_{k \mathop = 0}^\infty a_k x^k} = \sum_{k \mathop = 0}^\infty a_k$
where $\ds \lim_{x \mathop \to 1^-}$ denotes the [[Definition:Limit from... | Let $\epsilon > 0$.
Let $\ds \sum_{k \mathop = 0}^\infty a_k$ [[Definition:Convergent Series|converge]] to $s$.
Then its [[Definition:Sequence of Partial Sums|sequence of partial sums]] $\sequence {s_N}$, where $\ds s_N = \sum_{n \mathop = 1}^N a_n$, is a [[Convergent Sequence is Cauchy Sequence|Cauchy sequence]].
S... | Abel's Limit Theorem/Proof 1 | https://proofwiki.org/wiki/Abel's_Limit_Theorem | https://proofwiki.org/wiki/Abel's_Limit_Theorem/Proof_1 | [
"Abel's Limit Theorem",
"Series",
"Analysis",
"Abel's Theorem"
] | [
"Definition:Convergent Series",
"Definition:Series",
"Definition:Limit of Real Function/Left"
] | [
"Definition:Convergent Series",
"Definition:Series/Sequence of Partial Sums",
"Convergent Sequence is Cauchy Sequence",
"Abel's Lemma/Formulation 2",
"Sum of Geometric Sequence",
"Definition:Limit of Real Function/Left"
] |
proofwiki-1494 | Abel's Limit Theorem | Let $\ds \sum_{k \mathop = 0}^\infty a_k$ be a convergent series in $\R$.
Then:
:$\ds \lim_{x \mathop \to 1^-} \paren {\sum_{k \mathop = 0}^\infty a_k x^k} = \sum_{k \mathop = 0}^\infty a_k$
where $\ds \lim_{x \mathop \to 1^-}$ denotes the limit from the left. | {{tidy|Please attempt to understand the house style. Seriously, I haven't got the patience to tidy up.}}
{{MissingLinks}}
Since $\ds \sum_{k \mathop = 0}^\infty a_k$ converges and $\cmod{x^k}\le1$ and $\sequence{x^k}$ is decreasing,
by Abel's Test for Uniform Convergence $\ds \sum_{k \mathop = 0}^\infty a_kx^k$ converg... | Let $\ds \sum_{k \mathop = 0}^\infty a_k$ be a [[Definition:Convergent Series|convergent]] [[Definition:Series|series in $\R$]].
Then:
:$\ds \lim_{x \mathop \to 1^-} \paren {\sum_{k \mathop = 0}^\infty a_k x^k} = \sum_{k \mathop = 0}^\infty a_k$
where $\ds \lim_{x \mathop \to 1^-}$ denotes the [[Definition:Limit from... | {{tidy|Please attempt to understand the house style. Seriously, I haven't got the patience to tidy up.}}
{{MissingLinks}}
Since $\ds \sum_{k \mathop = 0}^\infty a_k$ converges and $\cmod{x^k}\le1$ and $\sequence{x^k}$ is decreasing,
by [[Abel's Test for Uniform Convergence]] $\ds \sum_{k \mathop = 0}^\infty a_kx^k$ co... | Abel's Limit Theorem/Proof 2 | https://proofwiki.org/wiki/Abel's_Limit_Theorem | https://proofwiki.org/wiki/Abel's_Limit_Theorem/Proof_2 | [
"Abel's Limit Theorem",
"Series",
"Analysis",
"Abel's Theorem"
] | [
"Definition:Convergent Series",
"Definition:Series",
"Definition:Limit of Real Function/Left"
] | [
"Abel's Test for Uniform Convergence",
"Uniform Limit Theorem"
] |
proofwiki-1495 | Alternating Series Test | Let $\sequence {a_n}_{N \mathop \ge 0}$ be a decreasing sequence of positive terms in $\R$ which converges with a limit of zero.
That is, let $\forall n \in \N: a_n \ge 0, a_{n + 1} \le a_n, a_n \to 0$ as $n \to \infty$
Then the series:
:$\ds \sum_{n \mathop = 1}^\infty \paren {-1}^{n - 1} a_n = a_1 - a_2 + a_3 - a_4 +... | First we show that for each $n > m$, we have $0 \le a_{m + 1} - a_{m + 2} + a_{m + 3} - \dotsb \pm a_n \le a_{m + 1}$. | Let $\sequence {a_n}_{N \mathop \ge 0}$ be a [[Definition:Decreasing Sequence|decreasing sequence]] of [[Definition:Positive Real Number|positive]] [[Definition:Term of Sequence|terms]] in $\R$ which [[Definition:Convergent Sequence|converges]] with a [[Definition:Limit of Sequence (Number Field)|limit]] of zero.
That... | First we show that for each $n > m$, we have $0 \le a_{m + 1} - a_{m + 2} + a_{m + 3} - \dotsb \pm a_n \le a_{m + 1}$. | Alternating Series Test | https://proofwiki.org/wiki/Alternating_Series_Test | https://proofwiki.org/wiki/Alternating_Series_Test | [
"Alternating Series Test",
"Alternating Series",
"Convergence Tests",
"Named Theorems"
] | [
"Definition:Decreasing/Sequence",
"Definition:Positive/Real Number",
"Definition:Term of Sequence",
"Definition:Convergent Sequence",
"Definition:Limit of Sequence (Number Field)",
"Definition:Convergent Series"
] | [] |
proofwiki-1496 | Euler's Identity | :$e^{i \pi} + 1 = 0$ | Follows directly from Euler's Formula $e^{i z} = \cos z + i \sin z$, by plugging in $z = \pi$:
:$e^{i \pi} + 1 = \cos \pi + i \sin \pi + 1 = -1 + i \times 0 + 1 = 0$
{{qed}} | :$e^{i \pi} + 1 = 0$ | Follows directly from [[Euler's Formula]] $e^{i z} = \cos z + i \sin z$, by plugging in $z = \pi$:
:$e^{i \pi} + 1 = \cos \pi + i \sin \pi + 1 = -1 + i \times 0 + 1 = 0$
{{qed}} | Euler's Identity | https://proofwiki.org/wiki/Euler's_Identity | https://proofwiki.org/wiki/Euler's_Identity | [
"Euler's Identity",
"Euler's Number",
"Pi",
"Complex Analysis"
] | [] | [
"Euler's Formula"
] |
proofwiki-1497 | Intermediate Value Theorem | Let $I \subseteq \R$ be a real interval.
Let $f: I \to \R$ be continuous on $I$.
Then $f$ is a Darboux function.
That is:
Let $a, b \in I$.
Let $k \in \R$ lie between $\map f a$ and $\map f b$.
That is, either:
:$\map f a < k < \map f b$
or:
:$\map f b < k < \map f a$
Then $\exists c \in \openint a b$ such that $\map f... | As the codomain of $f$ is $\closedint a b$, it follows that the image of $f$ is a subset of $\closedint a b$.
Thus:
:$\map f a \ge a$
and
:$\map f b \le b$
Let us define the real function $g: \closedint a b \to \R$ by:
:$\map g x = \map f x - x$
Then by the Combined Sum Rule for Continuous Real Functions, $\map g x$ is... | Let $I \subseteq \R$ be a [[Definition:Real Interval|real interval]].
Let $f: I \to \R$ be [[Definition:Continuous on Interval|continuous]] on $I$.
Then $f$ is a [[Definition:Darboux Function|Darboux function]].
That is:
Let $a, b \in I$.
Let $k \in \R$ lie between $\map f a$ and $\map f b$.
That is, either:
:... | As the [[Definition:Codomain of Mapping|codomain]] of $f$ is $\closedint a b$, it follows that the [[Image is Subset of Codomain|image of $f$ is a subset of $\closedint a b$]].
Thus:
:$\map f a \ge a$
and
:$\map f b \le b$
Let us define the [[Definition:Real Function|real function]] $g: \closedint a b \to \R$ by:
:$\... | Brouwer's Fixed Point Theorem/One-Dimensional Version/Proof by Intermediate Value Theorem | https://proofwiki.org/wiki/Intermediate_Value_Theorem | https://proofwiki.org/wiki/Brouwer's_Fixed_Point_Theorem/One-Dimensional_Version/Proof_by_Intermediate_Value_Theorem | [
"Intermediate Value Theorem",
"Darboux Functions",
"Real Analysis",
"Named Theorems"
] | [
"Definition:Real Interval",
"Definition:Continuous Real Function/Interval",
"Definition:Darboux Function"
] | [
"Definition:Codomain (Set Theory)/Mapping",
"Image is Subset of Codomain",
"Definition:Real Function",
"Combination Theorem for Continuous Functions/Real/Combined Sum Rule",
"Definition:Continuous Real Function/Interval",
"Intermediate Value Theorem"
] |
proofwiki-1498 | Intermediate Value Theorem | Let $I \subseteq \R$ be a real interval.
Let $f: I \to \R$ be continuous on $I$.
Then $f$ is a Darboux function.
That is:
Let $a, b \in I$.
Let $k \in \R$ lie between $\map f a$ and $\map f b$.
That is, either:
:$\map f a < k < \map f b$
or:
:$\map f b < k < \map f a$
Then $\exists c \in \openint a b$ such that $\map f... | This theorem is a restatement of Image of Real Interval under Continuous Real Function is Real Interval.
From Image of Real Interval under Continuous Real Function is Real Interval, the image of $\openint a b$ under $f$ is also a real interval (but not necessarily open).
Thus if $k$ lies between $\map f a$ and $\map f ... | Let $I \subseteq \R$ be a [[Definition:Real Interval|real interval]].
Let $f: I \to \R$ be [[Definition:Continuous on Interval|continuous]] on $I$.
Then $f$ is a [[Definition:Darboux Function|Darboux function]].
That is:
Let $a, b \in I$.
Let $k \in \R$ lie between $\map f a$ and $\map f b$.
That is, either:
:... | This theorem is a restatement of [[Image of Real Interval under Continuous Real Function is Real Interval]].
From [[Image of Real Interval under Continuous Real Function is Real Interval]], the [[Definition:Image of Subset under Mapping|image]] of $\openint a b$ under $f$ is also a [[Definition:Real Interval|real int... | Intermediate Value Theorem | https://proofwiki.org/wiki/Intermediate_Value_Theorem | https://proofwiki.org/wiki/Intermediate_Value_Theorem | [
"Intermediate Value Theorem",
"Darboux Functions",
"Real Analysis",
"Named Theorems"
] | [
"Definition:Real Interval",
"Definition:Continuous Real Function/Interval",
"Definition:Darboux Function"
] | [
"Image of Real Interval under Continuous Real Function is Real Interval",
"Image of Real Interval under Continuous Real Function is Real Interval",
"Definition:Image (Set Theory)/Mapping/Subset",
"Definition:Real Interval",
"Definition:Real Interval/Open"
] |
proofwiki-1499 | Tail of Convergent Series tends to Zero | Let $\sequence {a_n}_{n \mathop \ge 1}$ be a sequence of real numbers.
Let $\ds \sum_{n \mathop = 1}^\infty a_n$ be a convergent series.
Let $N \in \N_{\ge 1}$ be a natural number.
Let $\ds \sum_{n \mathop = N}^\infty a_n$ be the tail of the series $\ds \sum_{n \mathop = 1}^\infty a_n$.
Then:
:$\ds \sum_{n \mathop = N}... | Let $\sequence {s_n}$ be the sequence of partial sums of $\ds \sum_{n \mathop = 1}^\infty a_n$.
Let $\sequence {s'_n}$ be the sequence of partial sums of $\ds \sum_{n \mathop = N}^\infty a_n$.
It will be shown that $\sequence {s'_n}$ fulfils the Cauchy criterion.
That is:
:$\forall \epsilon \in \R_{>0}: \exists N: \for... | Let $\sequence {a_n}_{n \mathop \ge 1}$ be a [[Definition:Sequence|sequence]] of [[Definition:Real Numbers|real numbers]].
Let $\ds \sum_{n \mathop = 1}^\infty a_n$ be a [[Definition:Convergent Series|convergent series]].
Let $N \in \N_{\ge 1}$ be a [[Definition:Natural Numbers|natural number]].
Let $\ds \sum_{n \ma... | Let $\sequence {s_n}$ be the [[Definition:Sequence of Partial Sums|sequence of partial sums]] of $\ds \sum_{n \mathop = 1}^\infty a_n$.
Let $\sequence {s'_n}$ be the [[Definition:Sequence of Partial Sums|sequence of partial sums]] of $\ds \sum_{n \mathop = N}^\infty a_n$.
It will be shown that $\sequence {s'_n}$ ful... | Tail of Convergent Series tends to Zero | https://proofwiki.org/wiki/Tail_of_Convergent_Series_tends_to_Zero | https://proofwiki.org/wiki/Tail_of_Convergent_Series_tends_to_Zero | [
"Series"
] | [
"Definition:Sequence",
"Definition:Real Number",
"Definition:Convergent Series",
"Definition:Natural Numbers",
"Definition:Series/Tail",
"Definition:Series",
"Definition:Convergent Series",
"Definition:Series/Tail",
"Definition:Convergent Series"
] | [
"Definition:Series/Sequence of Partial Sums",
"Definition:Series/Sequence of Partial Sums",
"Definition:Cauchy Sequence/Cauchy Criterion",
"Definition:Strictly Positive/Real Number",
"Definition:Convergent Series",
"Definition:Cauchy Sequence/Cauchy Criterion",
"Convergent Sequence is Cauchy Sequence",
... |
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