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-14800 | Combination Theorem for Sequences/Complex/Multiple Rule | :$\ds \lim_{n \mathop \to \infty} \paren {\lambda z_n} = \lambda c$ | Let $\epsilon > 0$.
We need to find $N$ such that:
:$\forall n > N: \cmod {\lambda z_n - \lambda c} < \epsilon$
If $\lambda = 0$ the result is trivial.
So, assume $\lambda \ne 0$.
Then $\cmod \lambda > 0$ from the definition of the modulus of $\lambda$.
Hence $\dfrac \epsilon {\cmod \lambda} > 0$.
We have that $z_n \to... | :$\ds \lim_{n \mathop \to \infty} \paren {\lambda z_n} = \lambda c$ | Let $\epsilon > 0$.
We need to find $N$ such that:
:$\forall n > N: \cmod {\lambda z_n - \lambda c} < \epsilon$
If $\lambda = 0$ the result is trivial.
So, assume $\lambda \ne 0$.
Then $\cmod \lambda > 0$ from the definition of the [[Definition:Complex Modulus|modulus]] of $\lambda$.
Hence $\dfrac \epsilon {\cmod... | Combination Theorem for Sequences/Complex/Multiple Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Sequences/Complex/Multiple_Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Sequences/Complex/Multiple_Rule | [
"Combination Theorems for Sequences"
] | [] | [
"Definition:Complex Modulus",
"Complex Modulus of Product of Complex Numbers",
"Category:Combination Theorems for Sequences"
] |
proofwiki-14801 | Combination Theorem for Sequences/Complex/Combined Sum Rule | :$\ds \lim_{n \mathop \to \infty} \paren {\lambda z_n + \mu w_n} = \lambda c + \mu d$ | From the Multiple Rule for Complex Sequences, we have:
:$\ds \lim_{n \mathop \to \infty} \paren {\lambda z_n} = \lambda c$
:$\ds \lim_{n \mathop \to \infty} \paren {\mu w_n} = \mu d$
The result now follows directly from the Sum Rule for Complex Sequences:
:$\ds \lim_{n \mathop \to \infty} \paren {\lambda z_n + \mu w_n}... | :$\ds \lim_{n \mathop \to \infty} \paren {\lambda z_n + \mu w_n} = \lambda c + \mu d$ | From the [[Multiple Rule for Complex Sequences]], we have:
:$\ds \lim_{n \mathop \to \infty} \paren {\lambda z_n} = \lambda c$
:$\ds \lim_{n \mathop \to \infty} \paren {\mu w_n} = \mu d$
The result now follows directly from the [[Sum Rule for Complex Sequences]]:
:$\ds \lim_{n \mathop \to \infty} \paren {\lambda z_n +... | Combination Theorem for Sequences/Complex/Combined Sum Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Sequences/Complex/Combined_Sum_Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Sequences/Complex/Combined_Sum_Rule | [
"Combination Theorems for Sequences"
] | [] | [
"Combination Theorem for Sequences/Complex/Multiple Rule",
"Combination Theorem for Sequences/Complex/Sum Rule",
"Category:Combination Theorems for Sequences"
] |
proofwiki-14802 | Combination Theorem for Sequences/Normed Division Ring/Sum Rule | :$\sequence {x_n + y_n}$ is convergent and $\ds \lim_{n \mathop \to \infty} \paren {x_n + y_n} = l + m$ | Let $\epsilon > 0$ be given.
Then $\dfrac \epsilon 2 > 0$.
Since $\sequence {x_n}$ is convergent to $l$, we can find $N_1$ such that:
:$\forall n > N_1: \norm {x_n - l} < \dfrac \epsilon 2$
Similarly, for $\sequence {y_n}$ we can find $N_2$ such that:
:$\forall n > N_2: \norm {y_n - m} < \dfrac \epsilon 2$
Now let $N =... | :$\sequence {x_n + y_n}$ is [[Definition:Convergent Sequence in Normed Division Ring|convergent]] and $\ds \lim_{n \mathop \to \infty} \paren {x_n + y_n} = l + m$ | Let $\epsilon > 0$ be given.
Then $\dfrac \epsilon 2 > 0$.
Since $\sequence {x_n}$ is [[Definition:Convergent Sequence in Normed Division Ring|convergent]] to $l$, we can find $N_1$ such that:
:$\forall n > N_1: \norm {x_n - l} < \dfrac \epsilon 2$
Similarly, for $\sequence {y_n}$ we can find $N_2$ such that:
:$\for... | Combination Theorem for Sequences/Normed Division Ring/Sum Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Sequences/Normed_Division_Ring/Sum_Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Sequences/Normed_Division_Ring/Sum_Rule | [
"Combination Theorem for Sequences in Normed Division Rings"
] | [
"Definition:Convergent Sequence/Normed Division Ring"
] | [
"Definition:Convergent Sequence/Normed Division Ring",
"Definition:Convergent Sequence/Normed Division Ring",
"Category:Combination Theorem for Sequences in Normed Division Rings"
] |
proofwiki-14803 | Absolute Value Function is Completely Multiplicative | Let $x, y \in \R$ be real numbers.
Then:
:$\size {x y} = \size x \size y$
where $\size x$ denotes the absolute value of $x$.
Thus the absolute value function is completely multiplicative. | Let either $x = 0$ or $y = 0$, or both.
We have that $\size 0 = 0$ by definition of absolute value.
Hence:
:$\size x \size y = 0 = x y = \size {x y}$
Let $x > 0$ and $y > 0$.
Then:
{{begin-eqn}}
{{eqn | l = x y
| o = >
| r = 0
| c =
}}
{{eqn | ll= \leadsto
| l = \size {x y}
| r = x y
... | Let $x, y \in \R$ be [[Definition:Real Number|real numbers]].
Then:
:$\size {x y} = \size x \size y$
where $\size x$ denotes the [[Definition:Absolute Value|absolute value]] of $x$.
Thus the [[Definition:Absolute Value|absolute value function]] is [[Definition:Completely Multiplicative Function|completely multiplic... | Let either $x = 0$ or $y = 0$, or both.
We have that $\size 0 = 0$ by definition of [[Definition:Absolute Value|absolute value]].
Hence:
:$\size x \size y = 0 = x y = \size {x y}$
Let $x > 0$ and $y > 0$.
Then:
{{begin-eqn}}
{{eqn | l = x y
| o = >
| r = 0
| c =
}}
{{eqn | ll= \leadsto
| l... | Absolute Value Function is Completely Multiplicative/Proof 1 | https://proofwiki.org/wiki/Absolute_Value_Function_is_Completely_Multiplicative | https://proofwiki.org/wiki/Absolute_Value_Function_is_Completely_Multiplicative/Proof_1 | [
"Absolute Value Function is Completely Multiplicative",
"Absolute Value Function",
"Real Analysis",
"Completely Multiplicative Functions"
] | [
"Definition:Real Number",
"Definition:Absolute Value",
"Definition:Absolute Value",
"Definition:Completely Multiplicative Function"
] | [
"Definition:Absolute Value",
"Definition:Positive/Real Number",
"Definition:Negative/Real Number"
] |
proofwiki-14804 | Absolute Value Function is Completely Multiplicative | Let $x, y \in \R$ be real numbers.
Then:
:$\size {x y} = \size x \size y$
where $\size x$ denotes the absolute value of $x$.
Thus the absolute value function is completely multiplicative. | Let $x$ and $y$ be considered as complex numbers which are wholly real.
That is:
:$x = x + 0 i, y = y + 0 i$
From Complex Modulus of Real Number equals Absolute Value, the absolute value of $x$ and $y$ equal the complex moduli of $x + 0 i$ and $y + 0 i$.
Thus $\cmod x \cmod y$ can be interpreted as the complex modulus ... | Let $x, y \in \R$ be [[Definition:Real Number|real numbers]].
Then:
:$\size {x y} = \size x \size y$
where $\size x$ denotes the [[Definition:Absolute Value|absolute value]] of $x$.
Thus the [[Definition:Absolute Value|absolute value function]] is [[Definition:Completely Multiplicative Function|completely multiplic... | Let $x$ and $y$ be considered as [[Definition:Complex Number|complex numbers]] which are [[Definition:Wholly Real|wholly real]].
That is:
:$x = x + 0 i, y = y + 0 i$
From [[Complex Modulus of Real Number equals Absolute Value]], the [[Definition:Absolute Value|absolute value]] of $x$ and $y$ equal the [[Definition:Co... | Absolute Value Function is Completely Multiplicative/Proof 2 | https://proofwiki.org/wiki/Absolute_Value_Function_is_Completely_Multiplicative | https://proofwiki.org/wiki/Absolute_Value_Function_is_Completely_Multiplicative/Proof_2 | [
"Absolute Value Function is Completely Multiplicative",
"Absolute Value Function",
"Real Analysis",
"Completely Multiplicative Functions"
] | [
"Definition:Real Number",
"Definition:Absolute Value",
"Definition:Absolute Value",
"Definition:Completely Multiplicative Function"
] | [
"Definition:Complex Number",
"Definition:Complex Number/Wholly Real",
"Complex Modulus of Real Number equals Absolute Value",
"Definition:Absolute Value",
"Definition:Complex Modulus",
"Definition:Complex Modulus",
"Definition:Multiplication/Real Numbers",
"Definition:Complex Modulus",
"Complex Modu... |
proofwiki-14805 | Absolute Value Function is Completely Multiplicative | Let $x, y \in \R$ be real numbers.
Then:
:$\size {x y} = \size x \size y$
where $\size x$ denotes the absolute value of $x$.
Thus the absolute value function is completely multiplicative. | {{begin-eqn}}
{{eqn | l = \size {x y}
| r = \sqrt {\paren {x y}^2}
| c = {{Defof|Absolute Value|index = 2}}
}}
{{eqn | r = \sqrt {x^2 y^2}
| c =
}}
{{eqn | r = \sqrt {x^2} \sqrt{y^2}
| c =
}}
{{eqn | r = \size x \cdot \size y
| c = {{Defof|Absolute Value|index = 2}}
}}
{{end-eqn}}
{{qed}... | Let $x, y \in \R$ be [[Definition:Real Number|real numbers]].
Then:
:$\size {x y} = \size x \size y$
where $\size x$ denotes the [[Definition:Absolute Value|absolute value]] of $x$.
Thus the [[Definition:Absolute Value|absolute value function]] is [[Definition:Completely Multiplicative Function|completely multiplic... | {{begin-eqn}}
{{eqn | l = \size {x y}
| r = \sqrt {\paren {x y}^2}
| c = {{Defof|Absolute Value|index = 2}}
}}
{{eqn | r = \sqrt {x^2 y^2}
| c =
}}
{{eqn | r = \sqrt {x^2} \sqrt{y^2}
| c =
}}
{{eqn | r = \size x \cdot \size y
| c = {{Defof|Absolute Value|index = 2}}
}}
{{end-eqn}}
{{qed}... | Absolute Value Function is Completely Multiplicative/Proof 3 | https://proofwiki.org/wiki/Absolute_Value_Function_is_Completely_Multiplicative | https://proofwiki.org/wiki/Absolute_Value_Function_is_Completely_Multiplicative/Proof_3 | [
"Absolute Value Function is Completely Multiplicative",
"Absolute Value Function",
"Real Analysis",
"Completely Multiplicative Functions"
] | [
"Definition:Real Number",
"Definition:Absolute Value",
"Definition:Absolute Value",
"Definition:Completely Multiplicative Function"
] | [] |
proofwiki-14806 | Absolute Value Function is Completely Multiplicative | Let $x, y \in \R$ be real numbers.
Then:
:$\size {x y} = \size x \size y$
where $\size x$ denotes the absolute value of $x$.
Thus the absolute value function is completely multiplicative. | Follows directly from:
:Real Numbers form Ordered Integral Domain
:Product of Absolute Values on Ordered Integral Domain.
{{qed}} | Let $x, y \in \R$ be [[Definition:Real Number|real numbers]].
Then:
:$\size {x y} = \size x \size y$
where $\size x$ denotes the [[Definition:Absolute Value|absolute value]] of $x$.
Thus the [[Definition:Absolute Value|absolute value function]] is [[Definition:Completely Multiplicative Function|completely multiplic... | Follows directly from:
:[[Real Numbers form Ordered Integral Domain]]
:[[Product of Absolute Values on Ordered Integral Domain]].
{{qed}} | Absolute Value Function is Completely Multiplicative/Proof 4 | https://proofwiki.org/wiki/Absolute_Value_Function_is_Completely_Multiplicative | https://proofwiki.org/wiki/Absolute_Value_Function_is_Completely_Multiplicative/Proof_4 | [
"Absolute Value Function is Completely Multiplicative",
"Absolute Value Function",
"Real Analysis",
"Completely Multiplicative Functions"
] | [
"Definition:Real Number",
"Definition:Absolute Value",
"Definition:Absolute Value",
"Definition:Completely Multiplicative Function"
] | [
"Real Numbers form Ordered Integral Domain",
"Product of Absolute Values on Ordered Integral Domain"
] |
proofwiki-14807 | Combination Theorem for Sequences/Complex/Product Rule | :$\ds \lim_{n \mathop \to \infty} \paren {z_n w_n} = c d$ | Because $\sequence {z_n}$ converges, it is bounded by Convergent Sequence is Bounded.
Suppose $\cmod {z_n} \le K$ for $n = 1, 2, 3, \ldots$.
Then:
{{begin-eqn}}
{{eqn | l = \cmod {z_n w_n - c d}
| r = \cmod {z_n w_n - z_n d + z_n d - c d}
| c =
}}
{{eqn | o = \le
| r = \cmod {z_n w_n - z_n d} + \cmod... | :$\ds \lim_{n \mathop \to \infty} \paren {z_n w_n} = c d$ | Because $\sequence {z_n}$ [[Definition:Convergent Real Sequence|converges]], it is bounded by [[Convergent Sequence is Bounded]].
Suppose $\cmod {z_n} \le K$ for $n = 1, 2, 3, \ldots$.
Then:
{{begin-eqn}}
{{eqn | l = \cmod {z_n w_n - c d}
| r = \cmod {z_n w_n - z_n d + z_n d - c d}
| c =
}}
{{eqn | o = ... | Combination Theorem for Sequences/Complex/Product Rule/Proof 1 | https://proofwiki.org/wiki/Combination_Theorem_for_Sequences/Complex/Product_Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Sequences/Complex/Product_Rule/Proof_1 | [
"Combination Theorems for Sequences",
"Product Rule for Complex Sequences"
] | [] | [
"Definition:Convergent Sequence/Real Numbers",
"Convergent Sequence in Metric Space is Bounded",
"Triangle Inequality/Complex Numbers",
"Complex Modulus of Product of Complex Numbers",
"Convergent Sequence Minus Limit",
"Combination Theorem for Sequences/Real/Combined Sum Rule",
"Squeeze Theorem/Sequenc... |
proofwiki-14808 | Combination Theorem for Sequences/Complex/Product Rule | :$\ds \lim_{n \mathop \to \infty} \paren {z_n w_n} = c d$ | Let $z_n = x_n + i y_n$.
Let $w_n = u_n + i v_n$.
Let $c = a + i b$
Let $d = e + i f$.
By definition of convergent complex sequence:
{{begin-eqn}}
{{eqn | l = \lim_{n \mathop \to \infty} z_n
| r = c
| c =
}}
{{eqn | ll= \leadsto
| l = \lim_{n \mathop \to \infty} x_n + i \lim_{n \mathop \to \infty} y_... | :$\ds \lim_{n \mathop \to \infty} \paren {z_n w_n} = c d$ | Let $z_n = x_n + i y_n$.
Let $w_n = u_n + i v_n$.
Let $c = a + i b$
Let $d = e + i f$.
By definition of [[Definition:Convergent Complex Sequence|convergent complex sequence]]:
{{begin-eqn}}
{{eqn | l = \lim_{n \mathop \to \infty} z_n
| r = c
| c =
}}
{{eqn | ll= \leadsto
| l = \lim_{n \mathop \... | Combination Theorem for Sequences/Complex/Product Rule/Proof 2 | https://proofwiki.org/wiki/Combination_Theorem_for_Sequences/Complex/Product_Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Sequences/Complex/Product_Rule/Proof_2 | [
"Combination Theorems for Sequences",
"Product Rule for Complex Sequences"
] | [] | [
"Definition:Convergent Sequence/Complex Numbers",
"Combination Theorem for Sequences/Real/Sum Rule",
"Combination Theorem for Sequences/Real/Product Rule"
] |
proofwiki-14809 | Combination Theorem for Sequences/Complex/Quotient Rule | :$\ds \lim_{n \mathop \to \infty} \frac {z_n} {w_n} = \frac c d$
provided that $d \ne 0$. | As $z_n \to c$ as $n \to \infty$, it follows from Modulus of Limit that $\size {w_n} \to \size d$ as $n \to \infty$.
As $d \ne 0$, it follows from the definition of the modulus of $d$ that $\size d > 0$.
From Sequence Converges to Within Half Limit, we have $\exists N: \forall n > N: \size {w_n} > \dfrac {\size d} 2$.
... | :$\ds \lim_{n \mathop \to \infty} \frac {z_n} {w_n} = \frac c d$
provided that $d \ne 0$. | As $z_n \to c$ as $n \to \infty$, it follows from [[Modulus of Limit]] that $\size {w_n} \to \size d$ as $n \to \infty$.
As $d \ne 0$, it follows from the definition of the [[Definition:Complex Modulus|modulus]] of $d$ that $\size d > 0$.
From [[Sequence Converges to Within Half Limit]], we have $\exists N: \forall n... | Combination Theorem for Sequences/Complex/Quotient Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Sequences/Complex/Quotient_Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Sequences/Complex/Quotient_Rule | [
"Combination Theorems for Sequences"
] | [] | [
"Modulus of Limit",
"Definition:Complex Modulus",
"Sequence Converges to Within Half Limit",
"Squeeze Theorem/Sequences/Complex Numbers"
] |
proofwiki-14810 | Combination Theorem for Sequences/Normed Division Ring | Let $\struct {R, \norm {\, \cdot \,} }$ be a normed division ring.
Let $\sequence {x_n}$, $\sequence {y_n} $ be sequences in $R$.
Let $\sequence {x_n}$ and $\sequence {y_n}$ be convergent in the norm $\norm {\, \cdot \,}$ to the following limits:
:$\ds \lim_{n \mathop \to \infty} x_n = l$
:$\ds \lim_{n \mathop \to \in... | By Convergent Sequence in Normed Division Ring is Bounded, $\sequence {x_n}$ is bounded.
Suppose $\norm {x_n} \le K$ for $n = 1, 2, 3, \ldots$.
Let $M = \max \set {K, \norm m}$.
Then:
:$\norm m \le M$
and:
:$\forall n: \norm{x_n} \le M$
Let $\epsilon > 0$ be given.
Then $\dfrac \epsilon {2 M} > 0$.
As $\sequence {x_n}$... | Let $\struct {R, \norm {\, \cdot \,} }$ be a [[Definition:Normed Division Ring|normed division ring]].
Let $\sequence {x_n}$, $\sequence {y_n} $ be [[Definition:Sequence|sequences in $R$]].
Let $\sequence {x_n}$ and $\sequence {y_n}$ be [[Definition:Convergent Sequence in Normed Division Ring|convergent in the norm]... | By [[Convergent Sequence in Normed Division Ring is Bounded]], $\sequence {x_n}$ is [[Definition:Bounded Sequence in Normed Division Ring|bounded]].
Suppose $\norm {x_n} \le K$ for $n = 1, 2, 3, \ldots$.
Let $M = \max \set {K, \norm m}$.
Then:
:$\norm m \le M$
and:
:$\forall n: \norm{x_n} \le M$
Let $\epsilon > 0$... | Combination Theorem for Sequences/Normed Division Ring/Product Rule/Proof 1 | https://proofwiki.org/wiki/Combination_Theorem_for_Sequences/Normed_Division_Ring | https://proofwiki.org/wiki/Combination_Theorem_for_Sequences/Normed_Division_Ring/Product_Rule/Proof_1 | [
"Combination Theorem for Sequences in Normed Division Rings"
] | [
"Definition:Normed Division Ring",
"Definition:Sequence",
"Definition:Convergent Sequence/Normed Division Ring",
"Definition:Limit of Sequence/Normed Division Ring",
"Combination Theorem for Sequences/Normed Division Ring/Sum Rule",
"Combination Theorem for Sequences/Normed Division Ring/Difference Rule",... | [
"Convergent Sequence in Normed Division Ring is Bounded",
"Definition:Bounded Sequence/Normed Division Ring",
"Definition:Convergent Sequence/Normed Division Ring",
"Definition:Norm/Division Ring",
"Definition:Norm/Division Ring",
"Definition:Convergent Sequence/Normed Division Ring"
] |
proofwiki-14811 | Combination Theorem for Sequences/Normed Division Ring | Let $\struct {R, \norm {\, \cdot \,} }$ be a normed division ring.
Let $\sequence {x_n}$, $\sequence {y_n} $ be sequences in $R$.
Let $\sequence {x_n}$ and $\sequence {y_n}$ be convergent in the norm $\norm {\, \cdot \,}$ to the following limits:
:$\ds \lim_{n \mathop \to \infty} x_n = l$
:$\ds \lim_{n \mathop \to \in... | By Convergent Sequence in Normed Division Ring is Bounded, $\sequence {x_n}$ is bounded.
Suppose $\norm {x_n} \le K$ for $n = 1, 2, 3, \ldots$.
Then for $n = 1, 2, 3, \ldots$:
{{begin-eqn}}
{{eqn | l = \norm {x_n y_n - l m}
| r = \norm {x_n y_n - x_n m + x_n m - l m}
| c =
}}
{{eqn | o = \le
| r = \n... | Let $\struct {R, \norm {\, \cdot \,} }$ be a [[Definition:Normed Division Ring|normed division ring]].
Let $\sequence {x_n}$, $\sequence {y_n} $ be [[Definition:Sequence|sequences in $R$]].
Let $\sequence {x_n}$ and $\sequence {y_n}$ be [[Definition:Convergent Sequence in Normed Division Ring|convergent in the norm]... | By [[Convergent Sequence in Normed Division Ring is Bounded]], $\sequence {x_n}$ is [[Definition:Bounded Sequence in Normed Division Ring|bounded]].
Suppose $\norm {x_n} \le K$ for $n = 1, 2, 3, \ldots$.
Then for $n = 1, 2, 3, \ldots$:
{{begin-eqn}}
{{eqn | l = \norm {x_n y_n - l m}
| r = \norm {x_n y_n - x_n ... | Combination Theorem for Sequences/Normed Division Ring/Product Rule/Proof 2 | https://proofwiki.org/wiki/Combination_Theorem_for_Sequences/Normed_Division_Ring | https://proofwiki.org/wiki/Combination_Theorem_for_Sequences/Normed_Division_Ring/Product_Rule/Proof_2 | [
"Combination Theorem for Sequences in Normed Division Rings"
] | [
"Definition:Normed Division Ring",
"Definition:Sequence",
"Definition:Convergent Sequence/Normed Division Ring",
"Definition:Limit of Sequence/Normed Division Ring",
"Combination Theorem for Sequences/Normed Division Ring/Sum Rule",
"Combination Theorem for Sequences/Normed Division Ring/Difference Rule",... | [
"Convergent Sequence in Normed Division Ring is Bounded",
"Definition:Bounded Sequence/Normed Division Ring",
"Definition:Real Sequence",
"Definition:Convergent Sequence/Normed Division Ring",
"Combination Theorem for Sequences/Real/Combined Sum Rule",
"Squeeze Theorem/Sequences/Complex Numbers",
"Defin... |
proofwiki-14812 | Combination Theorem for Sequences/Normed Division Ring/Product Rule | :$\sequence {x_n y_n}$ is convergent to the limit $\ds \lim_{n \mathop \to \infty} \paren {x_n y_n} = l m$ | By Convergent Sequence in Normed Division Ring is Bounded, $\sequence {x_n}$ is bounded.
Suppose $\norm {x_n} \le K$ for $n = 1, 2, 3, \ldots$.
Let $M = \max \set {K, \norm m}$.
Then:
:$\norm m \le M$
and:
:$\forall n: \norm{x_n} \le M$
Let $\epsilon > 0$ be given.
Then $\dfrac \epsilon {2 M} > 0$.
As $\sequence {x_n}$... | :$\sequence {x_n y_n}$ is [[Definition:Convergent Sequence in Normed Division Ring|convergent]] to the [[Definition:Limit of Sequence (Normed Division Ring)|limit]] $\ds \lim_{n \mathop \to \infty} \paren {x_n y_n} = l m$ | By [[Convergent Sequence in Normed Division Ring is Bounded]], $\sequence {x_n}$ is [[Definition:Bounded Sequence in Normed Division Ring|bounded]].
Suppose $\norm {x_n} \le K$ for $n = 1, 2, 3, \ldots$.
Let $M = \max \set {K, \norm m}$.
Then:
:$\norm m \le M$
and:
:$\forall n: \norm{x_n} \le M$
Let $\epsilon > 0$... | Combination Theorem for Sequences/Normed Division Ring/Product Rule/Proof 1 | https://proofwiki.org/wiki/Combination_Theorem_for_Sequences/Normed_Division_Ring/Product_Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Sequences/Normed_Division_Ring/Product_Rule/Proof_1 | [
"Combination Theorem for Sequences in Normed Division Rings"
] | [
"Definition:Convergent Sequence/Normed Division Ring",
"Definition:Limit of Sequence/Normed Division Ring"
] | [
"Convergent Sequence in Normed Division Ring is Bounded",
"Definition:Bounded Sequence/Normed Division Ring",
"Definition:Convergent Sequence/Normed Division Ring",
"Definition:Norm/Division Ring",
"Definition:Norm/Division Ring",
"Definition:Convergent Sequence/Normed Division Ring"
] |
proofwiki-14813 | Combination Theorem for Sequences/Normed Division Ring/Product Rule | :$\sequence {x_n y_n}$ is convergent to the limit $\ds \lim_{n \mathop \to \infty} \paren {x_n y_n} = l m$ | By Convergent Sequence in Normed Division Ring is Bounded, $\sequence {x_n}$ is bounded.
Suppose $\norm {x_n} \le K$ for $n = 1, 2, 3, \ldots$.
Then for $n = 1, 2, 3, \ldots$:
{{begin-eqn}}
{{eqn | l = \norm {x_n y_n - l m}
| r = \norm {x_n y_n - x_n m + x_n m - l m}
| c =
}}
{{eqn | o = \le
| r = \n... | :$\sequence {x_n y_n}$ is [[Definition:Convergent Sequence in Normed Division Ring|convergent]] to the [[Definition:Limit of Sequence (Normed Division Ring)|limit]] $\ds \lim_{n \mathop \to \infty} \paren {x_n y_n} = l m$ | By [[Convergent Sequence in Normed Division Ring is Bounded]], $\sequence {x_n}$ is [[Definition:Bounded Sequence in Normed Division Ring|bounded]].
Suppose $\norm {x_n} \le K$ for $n = 1, 2, 3, \ldots$.
Then for $n = 1, 2, 3, \ldots$:
{{begin-eqn}}
{{eqn | l = \norm {x_n y_n - l m}
| r = \norm {x_n y_n - x_n ... | Combination Theorem for Sequences/Normed Division Ring/Product Rule/Proof 2 | https://proofwiki.org/wiki/Combination_Theorem_for_Sequences/Normed_Division_Ring/Product_Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Sequences/Normed_Division_Ring/Product_Rule/Proof_2 | [
"Combination Theorem for Sequences in Normed Division Rings"
] | [
"Definition:Convergent Sequence/Normed Division Ring",
"Definition:Limit of Sequence/Normed Division Ring"
] | [
"Convergent Sequence in Normed Division Ring is Bounded",
"Definition:Bounded Sequence/Normed Division Ring",
"Definition:Real Sequence",
"Definition:Convergent Sequence/Normed Division Ring",
"Combination Theorem for Sequences/Real/Combined Sum Rule",
"Squeeze Theorem/Sequences/Complex Numbers",
"Defin... |
proofwiki-14814 | Combination Theorem for Sequences/Normed Division Ring/Multiple Rule | :$\sequence {\lambda x_n}$ is convergent and $\ds \lim_{n \mathop \to \infty} \paren {\lambda x_n} = \lambda l$ | Let
:$\sequence {\tilde{x}_n} := \tuple {\lambda, \lambda, \lambda, \ldots}$
and:
:$\sequence {y_n} := \sequence {x_n}$
The claim follows from Product Rule for Sequences in Normed Division Ring, since:
:$\sequence {\lambda x_n} = \sequence {\tilde{x}_n y_n}$
{{qed}} | :$\sequence {\lambda x_n}$ is [[Definition:Convergent Sequence in Normed Division Ring|convergent]] and $\ds \lim_{n \mathop \to \infty} \paren {\lambda x_n} = \lambda l$ | Let
:$\sequence {\tilde{x}_n} := \tuple {\lambda, \lambda, \lambda, \ldots}$
and:
:$\sequence {y_n} := \sequence {x_n}$
The claim follows from [[Product Rule for Sequences in Normed Division Ring]], since:
:$\sequence {\lambda x_n} = \sequence {\tilde{x}_n y_n}$
{{qed}} | Combination Theorem for Sequences/Normed Division Ring/Multiple Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Sequences/Normed_Division_Ring/Multiple_Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Sequences/Normed_Division_Ring/Multiple_Rule | [
"Combination Theorem for Sequences in Normed Division Rings"
] | [
"Definition:Convergent Sequence/Normed Division Ring"
] | [
"Combination Theorem for Sequences/Normed Division Ring/Product Rule"
] |
proofwiki-14815 | Combination Theorem for Sequences/Normed Division Ring/Combined Sum Rule | :$\sequence {\lambda x_n + \mu y_n }$ is convergent and $\ds \lim_{n \mathop \to \infty} \paren {\lambda x_n + \mu y_n} = \lambda l + \mu m$ | From the Multiple Rule for Sequences in Normed Division Ring, we have:
:$\ds \lim_{n \mathop \to \infty} \paren {\lambda x_n} = \lambda l$
:$\ds \lim_{n \mathop \to \infty} \paren {\mu y_n} = \mu m$
The result now follows directly from the Sum Rule for Sequences in Normed Division Ring:
:$\ds \lim_{n \mathop \to \infty... | :$\sequence {\lambda x_n + \mu y_n }$ is [[Definition:Convergent Sequence in Normed Division Ring|convergent]] and $\ds \lim_{n \mathop \to \infty} \paren {\lambda x_n + \mu y_n} = \lambda l + \mu m$ | From the [[Multiple Rule for Sequences in Normed Division Ring]], we have:
:$\ds \lim_{n \mathop \to \infty} \paren {\lambda x_n} = \lambda l$
:$\ds \lim_{n \mathop \to \infty} \paren {\mu y_n} = \mu m$
The result now follows directly from the [[Sum Rule for Sequences in Normed Division Ring]]:
:$\ds \lim_{n \mathop \... | Combination Theorem for Sequences/Normed Division Ring/Combined Sum Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Sequences/Normed_Division_Ring/Combined_Sum_Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Sequences/Normed_Division_Ring/Combined_Sum_Rule | [
"Combination Theorem for Sequences in Normed Division Rings"
] | [
"Definition:Convergent Sequence/Normed Division Ring"
] | [
"Combination Theorem for Sequences/Normed Division Ring/Multiple Rule",
"Combination Theorem for Sequences/Normed Division Ring/Sum Rule"
] |
proofwiki-14816 | Combination Theorem for Sequences/Normed Division Ring/Difference Rule | :$\sequence {x_n - y_n}$ is convergent and $\ds \lim_{n \mathop \to \infty} \paren {x_n - y_n} = l - m$ | From Sum Rule for Sequences in Normed Division Ring:
:$\ds \lim_{n \mathop \to \infty} \paren {x_n + y_n} = l + m$
From Multiple Rule for Sequences in Normed Division Ring:
:$\ds \lim_{n \mathop \to \infty} \paren {-y_n} = -m$
Hence:
:$\ds \lim_{n \mathop \to \infty} \paren {x_n + \paren {-y_n} } = l + \paren {-m}$
The... | :$\sequence {x_n - y_n}$ is [[Definition:Convergent Sequence in Normed Division Ring|convergent]] and $\ds \lim_{n \mathop \to \infty} \paren {x_n - y_n} = l - m$ | From [[Sum Rule for Sequences in Normed Division Ring]]:
:$\ds \lim_{n \mathop \to \infty} \paren {x_n + y_n} = l + m$
From [[Multiple Rule for Sequences in Normed Division Ring]]:
:$\ds \lim_{n \mathop \to \infty} \paren {-y_n} = -m$
Hence:
:$\ds \lim_{n \mathop \to \infty} \paren {x_n + \paren {-y_n} } = l + \paren... | Combination Theorem for Sequences/Normed Division Ring/Difference Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Sequences/Normed_Division_Ring/Difference_Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Sequences/Normed_Division_Ring/Difference_Rule | [
"Combination Theorem for Sequences in Normed Division Rings"
] | [
"Definition:Convergent Sequence/Normed Division Ring"
] | [
"Combination Theorem for Sequences/Normed Division Ring/Sum Rule",
"Combination Theorem for Sequences/Normed Division Ring/Multiple Rule"
] |
proofwiki-14817 | Absolutely Convergent Series is Convergent/Complex Numbers | Let $\ds \sum_{n \mathop = 1}^\infty z_n$ be an absolutely convergent series in $\C$.
Then $\ds \sum_{n \mathop = 1}^\infty z_n$ is convergent. | Let $z_n = u_n + i v_n$.
We have that:
{{begin-eqn}}
{{eqn | l = \cmod {z_n}
| r = \sqrt { {u_n}^2 + {v_n}^2}
| c =
}}
{{eqn | o = >
| r = \sqrt { {u_n}^2}
| c =
}}
{{eqn | o = >
| r = \size {u_n}
| c =
}}
{{end-eqn}}
and similarly:
:$\cmod {z_n} > \size {v_n}$
From the Comparison... | Let $\ds \sum_{n \mathop = 1}^\infty z_n$ be an [[Definition:Absolutely Convergent Complex Series|absolutely convergent series in $\C$]].
Then $\ds \sum_{n \mathop = 1}^\infty z_n$ is [[Definition:Convergent Series of Numbers|convergent]]. | Let $z_n = u_n + i v_n$.
We have that:
{{begin-eqn}}
{{eqn | l = \cmod {z_n}
| r = \sqrt { {u_n}^2 + {v_n}^2}
| c =
}}
{{eqn | o = >
| r = \sqrt { {u_n}^2}
| c =
}}
{{eqn | o = >
| r = \size {u_n}
| c =
}}
{{end-eqn}}
and similarly:
:$\cmod {z_n} > \size {v_n}$
From the [[Com... | Absolutely Convergent Series is Convergent/Complex Numbers | https://proofwiki.org/wiki/Absolutely_Convergent_Series_is_Convergent/Complex_Numbers | https://proofwiki.org/wiki/Absolutely_Convergent_Series_is_Convergent/Complex_Numbers | [
"Absolutely Convergent Series is Convergent"
] | [
"Definition:Absolutely Convergent Series/Complex Numbers",
"Definition:Convergent Series/Number Field"
] | [
"Comparison Test",
"Definition:Series",
"Definition:Absolutely Convergent Series/Real Numbers",
"Absolutely Convergent Series is Convergent/Real Numbers",
"Definition:Convergent Series/Number Field",
"Convergence of Series of Complex Numbers by Real and Imaginary Part",
"Definition:Convergent Series/Num... |
proofwiki-14818 | Cauchy's Convergence Criterion for Series | A series $\ds \sum_{i \mathop = 0}^\infty a_i$ is convergent {{iff}} for every $\epsilon > 0$ there is a number $N \in \N$ such that:
:$\size {a_{n + 1} + a_{n + 2} + \cdots + a_m} < \epsilon$
holds for all $n \ge N$ and $m > n$.
{{explain|What domain is $\sequence {a_n}$ in?}} | Let:
:$\ds s_n = \sum_{i \mathop = 0}^n a_i$
Then $\sequence {s_n}$ is a sequence in $\R$.
From Cauchy's Convergence Criterion on Real Numbers, $\sequence {s_n}$ is convergent {{iff}} it is a Cauchy sequence.
For $m > n$ we have:
:$\size {s_m - s_n} = \size {a_{n + 1} + a_{n + 2} + \cdots + a_m}$
{{qed}}
Category:Cauc... | A series $\ds \sum_{i \mathop = 0}^\infty a_i$ is [[Definition:Convergent Sequence|convergent]] {{iff}} for every $\epsilon > 0$ there is a number $N \in \N$ such that:
:$\size {a_{n + 1} + a_{n + 2} + \cdots + a_m} < \epsilon$
holds for all $n \ge N$ and $m > n$.
{{explain|What domain is $\sequence {a_n}$ in?}} | Let:
:$\ds s_n = \sum_{i \mathop = 0}^n a_i$
Then $\sequence {s_n}$ is a [[Definition:Real Sequence|sequence in $\R$]].
From [[Cauchy's Convergence Criterion on Real Numbers]], $\sequence {s_n}$ is convergent {{iff}} it is a [[Definition:Real Cauchy Sequence|Cauchy sequence]].
For $m > n$ we have:
:$\size {s_m -... | Cauchy's Convergence Criterion for Series | https://proofwiki.org/wiki/Cauchy's_Convergence_Criterion_for_Series | https://proofwiki.org/wiki/Cauchy's_Convergence_Criterion_for_Series | [
"Cauchy's Convergence Criterion",
"Real Analysis"
] | [
"Definition:Convergent Sequence"
] | [
"Definition:Real Sequence",
"Cauchy's Convergence Criterion/Real Numbers",
"Definition:Cauchy Sequence/Real Numbers",
"Category:Cauchy's Convergence Criterion",
"Category:Real Analysis"
] |
proofwiki-14819 | Combination Theorem for Sequences/Normed Division Ring/Product Rule/Proof 1 | Let $\struct {R, \norm {\,\cdot\,}}$ be a normed division ring.
Let $\sequence {x_n}$, $\sequence {y_n} $ be sequences in $R$.
Let $\sequence {x_n}$ and $\sequence {y_n}$ be convergent in the norm $\norm {\,\cdot\,}$ to the following limits:
:$\ds \lim_{n \mathop \to \infty} x_n = l$
:$\ds \lim_{n \mathop \to \infty} ... | By Convergent Sequence in Normed Division Ring is Bounded, $\sequence {x_n}$ is bounded.
Suppose $\norm {x_n} \le K$ for $n = 1, 2, 3, \ldots$.
Let $M = \max \set {K, \norm m}$.
Then:
:$\norm m \le M$
and:
:$\forall n: \norm{x_n} \le M$
Let $\epsilon > 0$ be given.
Then $\dfrac \epsilon {2 M} > 0$.
As $\sequence {x_n}$... | Let $\struct {R, \norm {\,\cdot\,}}$ be a [[Definition:Normed Division Ring|normed division ring]].
Let $\sequence {x_n}$, $\sequence {y_n} $ be [[Definition:Sequence|sequences in $R$]].
Let $\sequence {x_n}$ and $\sequence {y_n}$ be [[Definition:Convergent Sequence in Normed Division Ring|convergent in the norm]] $... | By [[Convergent Sequence in Normed Division Ring is Bounded]], $\sequence {x_n}$ is [[Definition:Bounded Sequence in Normed Division Ring|bounded]].
Suppose $\norm {x_n} \le K$ for $n = 1, 2, 3, \ldots$.
Let $M = \max \set {K, \norm m}$.
Then:
:$\norm m \le M$
and:
:$\forall n: \norm{x_n} \le M$
Let $\epsilon > 0$... | Combination Theorem for Sequences/Normed Division Ring/Product Rule/Proof 1 | https://proofwiki.org/wiki/Combination_Theorem_for_Sequences/Normed_Division_Ring/Product_Rule/Proof_1 | https://proofwiki.org/wiki/Combination_Theorem_for_Sequences/Normed_Division_Ring/Product_Rule/Proof_1 | [
"Combination Theorem for Sequences in Normed Division Rings"
] | [
"Definition:Normed Division Ring",
"Definition:Sequence",
"Definition:Convergent Sequence/Normed Division Ring",
"Definition:Limit of Sequence/Normed Division Ring"
] | [
"Convergent Sequence in Normed Division Ring is Bounded",
"Definition:Bounded Sequence/Normed Division Ring",
"Definition:Convergent Sequence/Normed Division Ring",
"Definition:Norm/Division Ring",
"Definition:Norm/Division Ring",
"Definition:Convergent Sequence/Normed Division Ring"
] |
proofwiki-14820 | Combination Theorem for Sequences/Normed Division Ring/Product Rule/Proof 2 | Let $\struct {R, \norm {\,\cdot\,} }$ be a normed division ring.
Let $\sequence {x_n}$ and $\sequence {y_n}$ be sequences in $R$.
Let $\sequence {x_n}$ and $\sequence {y_n}$ be convergent in the norm $\norm {\,\cdot\,}$ to the following limits:
:$\ds \lim_{n \mathop \to \infty} x_n = l$
:$\ds \lim_{n \mathop \to \infty... | By Convergent Sequence in Normed Division Ring is Bounded, $\sequence {x_n}$ is bounded.
Suppose $\norm {x_n} \le K$ for $n = 1, 2, 3, \ldots$.
Then for $n = 1, 2, 3, \ldots$:
{{begin-eqn}}
{{eqn | l = \norm {x_n y_n - l m}
| r = \norm {x_n y_n - x_n m + x_n m - l m}
| c =
}}
{{eqn | o = \le
| r = \n... | Let $\struct {R, \norm {\,\cdot\,} }$ be a [[Definition:Normed Division Ring|normed division ring]].
Let $\sequence {x_n}$ and $\sequence {y_n}$ be [[Definition:Sequence|sequences in $R$]].
Let $\sequence {x_n}$ and $\sequence {y_n}$ be [[Definition:Convergent Sequence in Normed Division Ring|convergent in the norm]]... | By [[Convergent Sequence in Normed Division Ring is Bounded]], $\sequence {x_n}$ is [[Definition:Bounded Sequence in Normed Division Ring|bounded]].
Suppose $\norm {x_n} \le K$ for $n = 1, 2, 3, \ldots$.
Then for $n = 1, 2, 3, \ldots$:
{{begin-eqn}}
{{eqn | l = \norm {x_n y_n - l m}
| r = \norm {x_n y_n - x_n ... | Combination Theorem for Sequences/Normed Division Ring/Product Rule/Proof 2 | https://proofwiki.org/wiki/Combination_Theorem_for_Sequences/Normed_Division_Ring/Product_Rule/Proof_2 | https://proofwiki.org/wiki/Combination_Theorem_for_Sequences/Normed_Division_Ring/Product_Rule/Proof_2 | [
"Combination Theorem for Sequences in Normed Division Rings"
] | [
"Definition:Normed Division Ring",
"Definition:Sequence",
"Definition:Convergent Sequence/Normed Division Ring",
"Definition:Limit of Sequence/Normed Division Ring"
] | [
"Convergent Sequence in Normed Division Ring is Bounded",
"Definition:Bounded Sequence/Normed Division Ring",
"Definition:Real Sequence",
"Definition:Convergent Sequence/Normed Division Ring",
"Combination Theorem for Sequences/Real/Combined Sum Rule",
"Squeeze Theorem/Sequences/Complex Numbers",
"Defin... |
proofwiki-14821 | Convergent Sequence in Normed Division Ring is Bounded | Let $\struct {R, \norm {\,\cdot\,}}$ be a normed division ring.
Let $\sequence {x_n}$ be a sequence in $R$.
Let $\sequence {x_n}$ be convergent in the norm $\norm {\,\cdot\,}$ to the following limit:
:$\ds \lim_{n \mathop \to \infty} x_n = l$
Then $\sequence {x_n}$ is bounded. | Let $\sequence {x_n}$ be convergent in the norm $\norm {\,\cdot\,}$ to the limit $l$, then:
:$\forall \epsilon \in \R_{>0}: \exists N \in \N : \forall n \ge N: \norm {x_n - l} < \epsilon$
Let $n_1$ satisfy:
:$\forall n \ge n_1: \norm {x_n - l} < 1$
Then $\forall n \ge n_1$:
{{begin-eqn}}
{{eqn | l = \norm {x_n}
|... | Let $\struct {R, \norm {\,\cdot\,}}$ be a [[Definition:Normed Division Ring|normed division ring]].
Let $\sequence {x_n}$ be a [[Definition:Sequence|sequence in $R$]].
Let $\sequence {x_n}$ be [[Definition:Convergent Sequence in Normed Division Ring|convergent in the norm]] $\norm {\,\cdot\,}$ to the following [[Defi... | Let $\sequence {x_n}$ be [[Definition:Convergent Sequence in Normed Division Ring|convergent in the norm]] $\norm {\,\cdot\,}$ to the [[Definition:Limit of Sequence (Normed Division Ring)|limit]] $l$, then:
:$\forall \epsilon \in \R_{>0}: \exists N \in \N : \forall n \ge N: \norm {x_n - l} < \epsilon$
Let $n_1$ satis... | Convergent Sequence in Normed Division Ring is Bounded/Proof 1 | https://proofwiki.org/wiki/Convergent_Sequence_in_Normed_Division_Ring_is_Bounded | https://proofwiki.org/wiki/Convergent_Sequence_in_Normed_Division_Ring_is_Bounded/Proof_1 | [
"Sequences",
"Convergence",
"Normed Division Rings",
"Convergent Sequences in Normed Division Rings",
"Convergent Sequence in Normed Division Ring is Bounded"
] | [
"Definition:Normed Division Ring",
"Definition:Sequence",
"Definition:Convergent Sequence/Normed Division Ring",
"Definition:Limit of Sequence/Normed Division Ring",
"Definition:Bounded Sequence/Normed Division Ring"
] | [
"Definition:Convergent Sequence/Normed Division Ring",
"Definition:Limit of Sequence/Normed Division Ring",
"Definition:Bounded Sequence/Normed Division Ring"
] |
proofwiki-14822 | Convergent Sequence in Normed Division Ring is Bounded | Let $\struct {R, \norm {\,\cdot\,}}$ be a normed division ring.
Let $\sequence {x_n}$ be a sequence in $R$.
Let $\sequence {x_n}$ be convergent in the norm $\norm {\,\cdot\,}$ to the following limit:
:$\ds \lim_{n \mathop \to \infty} x_n = l$
Then $\sequence {x_n}$ is bounded. | Let $d$ be the metric induced on $R$ be the norm $\norm {\,\cdot\,}$.
Let $\sequence {x_n}$ be convergent to the limit $l$ in $\struct {R, \norm {\,\cdot\,}}$.
By the definition of a convergent sequence in a normed division ring, $\sequence {x_n} $ is convergent to the limit $l$ in $\struct {R, d}$.
By Convergent Seque... | Let $\struct {R, \norm {\,\cdot\,}}$ be a [[Definition:Normed Division Ring|normed division ring]].
Let $\sequence {x_n}$ be a [[Definition:Sequence|sequence in $R$]].
Let $\sequence {x_n}$ be [[Definition:Convergent Sequence in Normed Division Ring|convergent in the norm]] $\norm {\,\cdot\,}$ to the following [[Defi... | Let $d$ be the [[Definition:Metric Induced by Norm on Division Ring|metric induced]] on $R$ be the [[Definition:Norm on Division Ring|norm]] $\norm {\,\cdot\,}$.
Let $\sequence {x_n}$ be [[Definition:Convergent Sequence in Normed Division Ring|convergent]] to the [[Definition:Limit of Sequence (Normed Division Ring)|l... | Convergent Sequence in Normed Division Ring is Bounded/Proof 2 | https://proofwiki.org/wiki/Convergent_Sequence_in_Normed_Division_Ring_is_Bounded | https://proofwiki.org/wiki/Convergent_Sequence_in_Normed_Division_Ring_is_Bounded/Proof_2 | [
"Sequences",
"Convergence",
"Normed Division Rings",
"Convergent Sequences in Normed Division Rings",
"Convergent Sequence in Normed Division Ring is Bounded"
] | [
"Definition:Normed Division Ring",
"Definition:Sequence",
"Definition:Convergent Sequence/Normed Division Ring",
"Definition:Limit of Sequence/Normed Division Ring",
"Definition:Bounded Sequence/Normed Division Ring"
] | [
"Definition:Metric Induced by Norm on Division Ring",
"Definition:Norm/Division Ring",
"Definition:Convergent Sequence/Normed Division Ring",
"Definition:Limit of Sequence/Normed Division Ring",
"Definition:Convergent Sequence/Normed Division Ring",
"Definition:Convergent Sequence/Metric Space",
"Defini... |
proofwiki-14823 | Convergent Sequence in Normed Division Ring is Bounded | Let $\struct {R, \norm {\,\cdot\,}}$ be a normed division ring.
Let $\sequence {x_n}$ be a sequence in $R$.
Let $\sequence {x_n}$ be convergent in the norm $\norm {\,\cdot\,}$ to the following limit:
:$\ds \lim_{n \mathop \to \infty} x_n = l$
Then $\sequence {x_n}$ is bounded. | Let $\sequence {x_n}$ be convergent to the limit $l$ in $\struct {R, \norm {\,\cdot\,} }$.
By modulus of limit in normed division ring, $\sequence {\norm {x_n} }$ is a convergent sequence in $\R$.
By Convergent Real Sequence is Bounded, $\sequence {\norm {x_n} }$ is bounded.
That is:
:$\exists M \in \R_{> 0}: \forall n... | Let $\struct {R, \norm {\,\cdot\,}}$ be a [[Definition:Normed Division Ring|normed division ring]].
Let $\sequence {x_n}$ be a [[Definition:Sequence|sequence in $R$]].
Let $\sequence {x_n}$ be [[Definition:Convergent Sequence in Normed Division Ring|convergent in the norm]] $\norm {\,\cdot\,}$ to the following [[Defi... | Let $\sequence {x_n}$ be [[Definition:Convergent Sequence in Normed Division Ring|convergent]] to the [[Definition:Limit of Sequence (Normed Division Ring)|limit]] $l$ in $\struct {R, \norm {\,\cdot\,} }$.
By [[Modulus of Limit/Normed Division Ring|modulus of limit in normed division ring]], $\sequence {\norm {x_n} }$... | Convergent Sequence in Normed Division Ring is Bounded/Proof 3 | https://proofwiki.org/wiki/Convergent_Sequence_in_Normed_Division_Ring_is_Bounded | https://proofwiki.org/wiki/Convergent_Sequence_in_Normed_Division_Ring_is_Bounded/Proof_3 | [
"Sequences",
"Convergence",
"Normed Division Rings",
"Convergent Sequences in Normed Division Rings",
"Convergent Sequence in Normed Division Ring is Bounded"
] | [
"Definition:Normed Division Ring",
"Definition:Sequence",
"Definition:Convergent Sequence/Normed Division Ring",
"Definition:Limit of Sequence/Normed Division Ring",
"Definition:Bounded Sequence/Normed Division Ring"
] | [
"Definition:Convergent Sequence/Normed Division Ring",
"Definition:Limit of Sequence/Normed Division Ring",
"Modulus of Limit/Normed Division Ring",
"Definition:Convergent Sequence/Real Numbers",
"Convergent Real Sequence is Bounded",
"Definition:Bounded Sequence/Real",
"Definition:Bounded Sequence/Norm... |
proofwiki-14824 | Convergent Sequence in Normed Division Ring is Bounded | Let $\struct {R, \norm {\,\cdot\,}}$ be a normed division ring.
Let $\sequence {x_n}$ be a sequence in $R$.
Let $\sequence {x_n}$ be convergent in the norm $\norm {\,\cdot\,}$ to the following limit:
:$\ds \lim_{n \mathop \to \infty} x_n = l$
Then $\sequence {x_n}$ is bounded. | Let $\sequence {x_n}$ be convergent to the limit $l$ in $\struct {R, \norm {\,\cdot\,}}$.
By Convergent Sequence is Cauchy Sequence in Normed Division Ring, $\sequence {x_n}$ is a Cauchy sequence in $\struct {R, \norm {\,\cdot\,}}$.
By Cauchy Sequence in Normed Division Ring is Bounded, $\sequence {x_n}$ is a bounded s... | Let $\struct {R, \norm {\,\cdot\,}}$ be a [[Definition:Normed Division Ring|normed division ring]].
Let $\sequence {x_n}$ be a [[Definition:Sequence|sequence in $R$]].
Let $\sequence {x_n}$ be [[Definition:Convergent Sequence in Normed Division Ring|convergent in the norm]] $\norm {\,\cdot\,}$ to the following [[Defi... | Let $\sequence {x_n}$ be [[Definition:Convergent Sequence in Normed Division Ring|convergent]] to the [[Definition:Limit of Sequence (Normed Division Ring)|limit]] $l$ in $\struct {R, \norm {\,\cdot\,}}$.
By [[Convergent Sequence is Cauchy Sequence/Normed Division Ring|Convergent Sequence is Cauchy Sequence in Normed ... | Convergent Sequence in Normed Division Ring is Bounded/Proof 4 | https://proofwiki.org/wiki/Convergent_Sequence_in_Normed_Division_Ring_is_Bounded | https://proofwiki.org/wiki/Convergent_Sequence_in_Normed_Division_Ring_is_Bounded/Proof_4 | [
"Sequences",
"Convergence",
"Normed Division Rings",
"Convergent Sequences in Normed Division Rings",
"Convergent Sequence in Normed Division Ring is Bounded"
] | [
"Definition:Normed Division Ring",
"Definition:Sequence",
"Definition:Convergent Sequence/Normed Division Ring",
"Definition:Limit of Sequence/Normed Division Ring",
"Definition:Bounded Sequence/Normed Division Ring"
] | [
"Definition:Convergent Sequence/Normed Division Ring",
"Definition:Limit of Sequence/Normed Division Ring",
"Convergent Sequence is Cauchy Sequence/Normed Division Ring",
"Definition:Cauchy Sequence/Normed Division Ring",
"Cauchy Sequence is Bounded/Normed Division Ring",
"Definition:Bounded Sequence/Norm... |
proofwiki-14825 | Metric Induced by Norm on Normed Division Ring is Metric | Let $\struct {R, \norm {\,\cdot\,} }$ be a normed division ring with zero and unity denoted by $0_R$ and $1_R$ respectively.
Let $d$ be the metric induced by $\norm{\,\cdot\,}$.
Then $d$ is a metric. | === Proof of {{Metric-space-axiom|1|nolink}} and {{Metric-space-axiom|4|nolink}} ===
Let $x, y \in R$.
Then $\map d {x, y} = \norm {x - y} \ge 0$, and furthermore:
{{begin-eqn}}
{{eqn | l = \map d {x, y}
| r = 0
}}
{{eqn | ll= \leadstoandfrom
| l = \norm {x - y}
| r = 0
| c = {{Defof|Metric Ind... | Let $\struct {R, \norm {\,\cdot\,} }$ be a [[Definition:Normed Division Ring|normed division ring]] with [[Definition:Ring Zero|zero]] and [[Definition:Unity of Ring|unity]] denoted by $0_R$ and $1_R$ respectively.
Let $d$ be the [[Definition:Metric Induced by Norm on Division Ring|metric induced by $\norm{\,\cdot\,}$... | === Proof of {{Metric-space-axiom|1|nolink}} and {{Metric-space-axiom|4|nolink}} ===
Let $x, y \in R$.
Then $\map d {x, y} = \norm {x - y} \ge 0$, and furthermore:
{{begin-eqn}}
{{eqn | l = \map d {x, y}
| r = 0
}}
{{eqn | ll= \leadstoandfrom
| l = \norm {x - y}
| r = 0
| c = {{Defof|Metric ... | Metric Induced by Norm on Normed Division Ring is Metric | https://proofwiki.org/wiki/Metric_Induced_by_Norm_on_Normed_Division_Ring_is_Metric | https://proofwiki.org/wiki/Metric_Induced_by_Norm_on_Normed_Division_Ring_is_Metric | [
"Normed Division Rings",
"Metric Spaces"
] | [
"Definition:Normed Division Ring",
"Definition:Ring Zero",
"Definition:Unity (Abstract Algebra)/Ring",
"Definition:Metric Induced by Norm on Division Ring",
"Definition:Metric Space/Metric"
] | [] |
proofwiki-14826 | Existence of Radius of Convergence of Complex Power Series/Absolute Convergence | Let $\map {B_R} \xi$ denote the open $R$-ball of $\xi$.
Let $z \in \map {B_R} \xi$.
Then $\map S z$ converges absolutely.
If $R = +\infty$, we define $\map {B_R} \xi = \C$. | Let $z \in \map {B_R} \xi$.
By definition of the open $R$-ball of $\xi$:
:$\cmod {z - \xi} < R$
where $\cmod z$ denotes the complex modulus of $z$.
By definition of radius of convergence, it follows that $\map S z$ converges.
Suppose $R$ is finite.
Let $\epsilon = R - \cmod {z - \xi} > 0$.
Now, let $w \in \map {B_R} \x... | Let $\map {B_R} \xi$ denote the [[Definition:Open Ball|open $R$-ball]] of $\xi$.
Let $z \in \map {B_R} \xi$.
Then $\map S z$ [[Definition:Absolutely Convergent Series|converges absolutely]].
If $R = +\infty$, we define $\map {B_R} \xi = \C$. | Let $z \in \map {B_R} \xi$.
By definition of the [[Definition:Open Ball|open $R$-ball]] of $\xi$:
:$\cmod {z - \xi} < R$
where $\cmod z$ denotes the [[Definition:Complex Modulus|complex modulus]] of $z$.
By definition of [[Definition:Radius of Convergence of Complex Power Series|radius of convergence]], it follows th... | Existence of Radius of Convergence of Complex Power Series/Absolute Convergence | https://proofwiki.org/wiki/Existence_of_Radius_of_Convergence_of_Complex_Power_Series/Absolute_Convergence | https://proofwiki.org/wiki/Existence_of_Radius_of_Convergence_of_Complex_Power_Series/Absolute_Convergence | [
"Complex Power Series",
"Convergence"
] | [
"Definition:Open Ball",
"Definition:Absolutely Convergent Series"
] | [
"Definition:Open Ball",
"Definition:Complex Modulus",
"Definition:Radius of Convergence/Complex Domain",
"Definition:Convergent Series",
"Definition:Complex Number",
"Definition:Complex Number",
"Nth Root Test",
"Definition:Divergent Series",
"Nth Root Test",
"Definition:Absolutely Convergent Seri... |
proofwiki-14827 | Existence of Radius of Convergence of Complex Power Series/Divergence | Let $\map { {B_R}^-} \xi$ denote the closed $R$-ball of $\xi$.
Let $z \notin \map { {B_R}^-} \xi$.
Then $\map S z$ is divergent. | Let $z \notin \map { {B_R}^-} \xi$.
Then by definition of the closed $R$-ball of $\xi$:
:$\cmod {z - \xi} > R$
where $\cmod z$ denotes the complex modulus of $z$.
By definition of radius of convergence, there exists $w \in \C$ such that:
:$\cmod {w - \xi} < \cmod {z - \xi}$
and $S \paren w$ is divergent.
Then:
{{begin-... | Let $\map { {B_R}^-} \xi$ denote the [[Definition:Closed Ball|closed $R$-ball]] of $\xi$.
Let $z \notin \map { {B_R}^-} \xi$.
Then $\map S z$ is [[Definition:Divergent Series|divergent]]. | Let $z \notin \map { {B_R}^-} \xi$.
Then by definition of the [[Definition:Closed Ball|closed $R$-ball]] of $\xi$:
:$\cmod {z - \xi} > R$
where $\cmod z$ denotes the [[Definition:Complex Modulus|complex modulus]] of $z$.
By definition of [[Definition:Radius of Convergence of Complex Power Series|radius of convergenc... | Existence of Radius of Convergence of Complex Power Series/Divergence | https://proofwiki.org/wiki/Existence_of_Radius_of_Convergence_of_Complex_Power_Series/Divergence | https://proofwiki.org/wiki/Existence_of_Radius_of_Convergence_of_Complex_Power_Series/Divergence | [
"Complex Power Series",
"Convergence"
] | [
"Definition:Closed Ball",
"Definition:Divergent Series"
] | [
"Definition:Closed Ball",
"Definition:Complex Modulus",
"Definition:Radius of Convergence/Complex Domain",
"Definition:Divergent Series",
"Nth Root Test",
"Definition:Divergent Series",
"Nth Root Test",
"Definition:Divergent Series"
] |
proofwiki-14828 | Radius of Convergence of Power Series in Complex Plane | Consider the complex power series:
:$S = \ds \sum_{k \mathop = 0}^\infty z^n$
The radius of convergence $S$ is $1$. | {{begin-eqn}}
{{eqn | l = \lim_{n \mathop \to \infty} \dfrac {\cmod {z^n} } {\cmod {z^{n - 1} } }
| r = \lim_{n \mathop \to \infty} \cmod {\dfrac {z^n} {z^{n - 1} } }
| c =
}}
{{eqn | r = \lim_{n \mathop \to \infty} \cmod z
| c =
}}
{{eqn | r = \cmod z
| c =
}}
{{end-eqn}}
By the Ratio Test, ... | Consider the [[Definition:Complex Power Series|complex power series]]:
:$S = \ds \sum_{k \mathop = 0}^\infty z^n$
The [[Definition:Radius of Convergence of Complex Power Series|radius of convergence]] $S$ is $1$. | {{begin-eqn}}
{{eqn | l = \lim_{n \mathop \to \infty} \dfrac {\cmod {z^n} } {\cmod {z^{n - 1} } }
| r = \lim_{n \mathop \to \infty} \cmod {\dfrac {z^n} {z^{n - 1} } }
| c =
}}
{{eqn | r = \lim_{n \mathop \to \infty} \cmod z
| c =
}}
{{eqn | r = \cmod z
| c =
}}
{{end-eqn}}
By the [[Ratio Tes... | Radius of Convergence of Power Series in Complex Plane | https://proofwiki.org/wiki/Radius_of_Convergence_of_Power_Series_in_Complex_Plane | https://proofwiki.org/wiki/Radius_of_Convergence_of_Power_Series_in_Complex_Plane | [
"Complex Power Series",
"Radius of Convergence"
] | [
"Definition:Power Series/Complex Domain",
"Definition:Radius of Convergence/Complex Domain"
] | [
"Ratio Test",
"Definition:Convergent Series/Number Field",
"Definition:Divergent Series",
"Definition:Radius of Convergence/Complex Domain"
] |
proofwiki-14829 | Radius of Convergence of Power Series Expansion for Cosine Function | The cosine function has the complex power series expansion:
{{begin-eqn}}
{{eqn | l = \map C z
| r = \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {z^{2 n} } {\paren {2 n}!}
| c =
}}
{{eqn | r = 1 - \frac {z^2} {2!} + \frac {z^4} {4!} - \frac {z^6} {6!} + \cdots
| c =
}}
{{end-eqn}}
which is the p... | Applying Radius of Convergence from Limit of Sequence: Complex Case, we find that:
{{begin-eqn}}
{{eqn | l = \lim_{n \mathop \to \infty} \cmod {\dfrac {a_{n + 1} } {a_n} }
| r = \lim_{n \mathop \to \infty} \cmod {\dfrac {\frac {\paren {-1}^{n + 1} } {\paren {2 \paren {n + 1} }!} } {\frac {\paren {-1}^n} {\paren {... | The [[Definition:Cosine Function|cosine function]] has the [[Definition:Complex Power Series|complex power series expansion]]:
{{begin-eqn}}
{{eqn | l = \map C z
| r = \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {z^{2 n} } {\paren {2 n}!}
| c =
}}
{{eqn | r = 1 - \frac {z^2} {2!} + \frac {z^4} {4!} - ... | Applying [[Radius of Convergence from Limit of Sequence/Complex Case|Radius of Convergence from Limit of Sequence: Complex Case]], we find that:
{{begin-eqn}}
{{eqn | l = \lim_{n \mathop \to \infty} \cmod {\dfrac {a_{n + 1} } {a_n} }
| r = \lim_{n \mathop \to \infty} \cmod {\dfrac {\frac {\paren {-1}^{n + 1} } {... | Radius of Convergence of Power Series Expansion for Cosine Function | https://proofwiki.org/wiki/Radius_of_Convergence_of_Power_Series_Expansion_for_Cosine_Function | https://proofwiki.org/wiki/Radius_of_Convergence_of_Power_Series_Expansion_for_Cosine_Function | [
"Radius of Convergence"
] | [
"Definition:Cosine",
"Definition:Power Series/Complex Domain",
"Definition:Power Series/Complex Domain",
"Definition:Cosine"
] | [
"Radius of Convergence from Limit of Sequence/Complex Case",
"Sequence of Powers of Reciprocals is Null Sequence"
] |
proofwiki-14830 | Radius of Convergence of Power Series Expansion for Sine Function | The sine function has the complex power series expansion:
{{begin-eqn}}
{{eqn | l = S \paren z
| r = \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {z^{2 n + 1} } {\paren {2 n + 1}!}
| c =
}}
{{eqn | r = z - \frac {z^3} {3!} + \frac {z^5} {5!} - \frac {z^7} {7!} + \cdots
| c =
}}
{{end-eqn}}
which ... | Applying Radius of Convergence from Limit of Sequence: Complex Case, we find that:
{{begin-eqn}}
{{eqn | l = \lim_{n \mathop \to \infty} \cmod {\dfrac {a_{n + 1} } {a_n} }
| r = \lim_{n \mathop \to \infty} \cmod {\dfrac {\frac {\paren {-1}^{n + 1} } {\paren {2 \paren {n + 1} + 1}!} } {\frac {\paren {-1}^n} {\pare... | The [[Definition:Sine Function|sine function]] has the [[Definition:Complex Power Series|complex power series expansion]]:
{{begin-eqn}}
{{eqn | l = S \paren z
| r = \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {z^{2 n + 1} } {\paren {2 n + 1}!}
| c =
}}
{{eqn | r = z - \frac {z^3} {3!} + \frac {z^5} {... | Applying [[Radius of Convergence from Limit of Sequence/Complex Case|Radius of Convergence from Limit of Sequence: Complex Case]], we find that:
{{begin-eqn}}
{{eqn | l = \lim_{n \mathop \to \infty} \cmod {\dfrac {a_{n + 1} } {a_n} }
| r = \lim_{n \mathop \to \infty} \cmod {\dfrac {\frac {\paren {-1}^{n + 1} } {... | Radius of Convergence of Power Series Expansion for Sine Function | https://proofwiki.org/wiki/Radius_of_Convergence_of_Power_Series_Expansion_for_Sine_Function | https://proofwiki.org/wiki/Radius_of_Convergence_of_Power_Series_Expansion_for_Sine_Function | [
"Radius of Convergence"
] | [
"Definition:Sine",
"Definition:Power Series/Complex Domain",
"Definition:Power Series/Complex Domain",
"Definition:Sine"
] | [
"Radius of Convergence from Limit of Sequence/Complex Case",
"Sequence of Powers of Reciprocals is Null Sequence"
] |
proofwiki-14831 | Sum of Infinite Series of Product of nth Power of Cosine by nth Multiple of Cosine | Let $0 < \theta < \dfrac \pi 2$.
Then:
{{begin-eqn}}
{{eqn | l = \sum_{n \mathop = 0}^\infty \cos^n \theta \, \map \cos {n + 1} \theta
| r = \cos \theta + \cos \theta \cos 2 \theta + \cos^2 \theta \cos 3 \theta + \cos^3 \theta \cos 4 \theta + \cdots
| c =
}}
{{eqn | r = 0
| c =
}}
{{end-eqn}} | Let $0 < \theta < \dfrac \pi 2$.
Then $0 < \cos \theta < 1$.
{{begin-eqn}}
{{eqn | l = \sum_{k \mathop = 0}^\infty r^k \cos k \theta
| r = \dfrac {1 - r \cos \theta} {1 - 2 r \cos \theta + r^2}
| c = Sum of Infinite Series of Product of Power and Cosine: $\size r < 1$
}}
{{eqn | ll= \leadsto
| l = \su... | Let $0 < \theta < \dfrac \pi 2$.
Then:
{{begin-eqn}}
{{eqn | l = \sum_{n \mathop = 0}^\infty \cos^n \theta \, \map \cos {n + 1} \theta
| r = \cos \theta + \cos \theta \cos 2 \theta + \cos^2 \theta \cos 3 \theta + \cos^3 \theta \cos 4 \theta + \cdots
| c =
}}
{{eqn | r = 0
| c =
}}
{{end-eqn}} | Let $0 < \theta < \dfrac \pi 2$.
Then $0 < \cos \theta < 1$.
{{begin-eqn}}
{{eqn | l = \sum_{k \mathop = 0}^\infty r^k \cos k \theta
| r = \dfrac {1 - r \cos \theta} {1 - 2 r \cos \theta + r^2}
| c = [[Sum of Infinite Series of Product of Power and Cosine]]: $\size r < 1$
}}
{{eqn | ll= \leadsto
| l... | Sum of Infinite Series of Product of nth Power of Cosine by nth Multiple of Cosine | https://proofwiki.org/wiki/Sum_of_Infinite_Series_of_Product_of_nth_Power_of_Cosine_by_nth_Multiple_of_Cosine | https://proofwiki.org/wiki/Sum_of_Infinite_Series_of_Product_of_nth_Power_of_Cosine_by_nth_Multiple_of_Cosine | [
"Cosine Function"
] | [] | [
"Sum of Infinite Series of Product of Power and Cosine",
"Translation of Index Variable of Summation"
] |
proofwiki-14832 | Euler's Formula/Real Domain | Let $\theta \in \R$ be a real number.
Then:
:$e^{i \theta} = \cos \theta + i \sin \theta$ | Consider the differential equation:
:$D_z \map f z = i \cdot \map f z$
=== Step 1 ===
We will prove that $z = \cos \theta + i \sin \theta$ is a solution.
{{begin-eqn}}
{{eqn | l = z
| r = \cos \theta + i \sin \theta
| c =
}}
{{eqn | l = \frac {\d z} {\d \theta}
| r = -\sin \theta + i \cos \theta
... | Let $\theta \in \R$ be a [[Definition:Real Number|real number]].
Then:
:$e^{i \theta} = \cos \theta + i \sin \theta$ | Consider the [[Definition:Differential Equation|differential equation]]:
:$D_z \map f z = i \cdot \map f z$
=== Step 1 ===
We will prove that $z = \cos \theta + i \sin \theta$ is a [[Definition:Solution to Differential Equation|solution]].
{{begin-eqn}}
{{eqn | l = z
| r = \cos \theta + i \sin \theta
|... | Euler's Formula/Real Domain/Proof 1 | https://proofwiki.org/wiki/Euler's_Formula/Real_Domain | https://proofwiki.org/wiki/Euler's_Formula/Real_Domain/Proof_1 | [
"Euler's Formula",
"Exponential Function",
"Trigonometric Functions"
] | [
"Definition:Real Number"
] | [
"Definition:Differential Equation",
"Definition:Differential Equation/Solution",
"Derivative of Sine Function",
"Derivative of Cosine Function",
"Linear Combination of Derivatives",
"Definition:Differential Equation/Solution",
"Derivative of Exponential Function",
"Derivative of Composite Function",
... |
proofwiki-14833 | Euler's Formula/Real Domain | Let $\theta \in \R$ be a real number.
Then:
:$e^{i \theta} = \cos \theta + i \sin \theta$ | This:
:$e^{i \theta} = \cos \theta + i \sin \theta$
is logically equivalent to this:
:$\dfrac {\cos \theta + i \sin \theta} {e^{i \theta} } = 1$
for every $\theta$.
Note that the left expression is nowhere undefined.
Taking the derivative of this:
{{begin-eqn}}
{{eqn | l = \dfrac \d {\d \theta} e^{-i \theta} \paren {\c... | Let $\theta \in \R$ be a [[Definition:Real Number|real number]].
Then:
:$e^{i \theta} = \cos \theta + i \sin \theta$ | This:
:$e^{i \theta} = \cos \theta + i \sin \theta$
is [[Definition:Logical Equivalence|logically equivalent]] to this:
:$\dfrac {\cos \theta + i \sin \theta} {e^{i \theta} } = 1$
for every $\theta$.
Note that the left expression is nowhere undefined.
Taking the [[Definition:Derivative of Complex Function|deriva... | Euler's Formula/Real Domain/Proof 2 | https://proofwiki.org/wiki/Euler's_Formula/Real_Domain | https://proofwiki.org/wiki/Euler's_Formula/Real_Domain/Proof_2 | [
"Euler's Formula",
"Exponential Function",
"Trigonometric Functions"
] | [
"Definition:Real Number"
] | [
"Definition:Logical Equivalence",
"Definition:Derivative/Complex Function",
"Product Rule for Derivatives",
"Derivative of Exponential Function",
"Definition:Constant Mapping"
] |
proofwiki-14834 | Euler's Formula/Real Domain | Let $\theta \in \R$ be a real number.
Then:
:$e^{i \theta} = \cos \theta + i \sin \theta$ | It follows from Argument of Product equals Sum of Arguments that the $\map \arg z$ function for all $z \in \C$ satisfies the relationship:
:$\map \arg {z_1 z_2} = \map \arg {z_1} + \map \arg {z_2}$
which means that $\map \arg z$ is a kind of logarithm, in the sense that it satisfies the fundamental property of logarith... | Let $\theta \in \R$ be a [[Definition:Real Number|real number]].
Then:
:$e^{i \theta} = \cos \theta + i \sin \theta$ | It follows from [[Argument of Product equals Sum of Arguments]] that the [[Definition:Argument of Complex Number|$\map \arg z$ function]] for all $z \in \C$ satisfies the relationship:
:$\map \arg {z_1 z_2} = \map \arg {z_1} + \map \arg {z_2}$
which means that $\map \arg z$ is a kind of [[Definition:General Logarithm... | Euler's Formula/Real Domain/Proof 3 | https://proofwiki.org/wiki/Euler's_Formula/Real_Domain | https://proofwiki.org/wiki/Euler's_Formula/Real_Domain/Proof_3 | [
"Euler's Formula",
"Exponential Function",
"Trigonometric Functions"
] | [
"Definition:Real Number"
] | [
"Argument of Product equals Sum of Arguments",
"Definition:Argument of Complex Number",
"Definition:General Logarithm",
"Definition:Real Function",
"Definition:Complex Modulus",
"Definition:Complex Function",
"Definition:Complex Number",
"Derivative of Composite Function",
"Definition:Inverse Mappin... |
proofwiki-14835 | Euler's Formula/Real Domain | Let $\theta \in \R$ be a real number.
Then:
:$e^{i \theta} = \cos \theta + i \sin \theta$ | Note that the following proof, as written, only holds for real $\theta$.
Define:
:$\map x \theta = e^{i \theta}$
:$\map y \theta = \cos \theta + i \sin \theta$
Consider first $\theta \ge 0$.
Taking Laplace transforms:
{{begin-eqn}}
{{eqn | l = \map {\laptrans {\map x \theta} } s
| r = \map {\laptrans {e^{i \theta... | Let $\theta \in \R$ be a [[Definition:Real Number|real number]].
Then:
:$e^{i \theta} = \cos \theta + i \sin \theta$ | Note that the following proof, as written, only holds for [[Definition:Real Number|real]] $\theta$.
Define:
:$\map x \theta = e^{i \theta}$
:$\map y \theta = \cos \theta + i \sin \theta$
Consider first $\theta \ge 0$.
Taking [[Definition:Laplace Transform|Laplace transforms]]:
{{begin-eqn}}
{{eqn | l = \map {\lap... | Euler's Formula/Real Domain/Proof 4 | https://proofwiki.org/wiki/Euler's_Formula/Real_Domain | https://proofwiki.org/wiki/Euler's_Formula/Real_Domain/Proof_4 | [
"Euler's Formula",
"Exponential Function",
"Trigonometric Functions"
] | [
"Definition:Real Number"
] | [
"Definition:Real Number",
"Definition:Laplace Transform",
"Laplace Transform of Exponential",
"Linear Combination of Laplace Transforms",
"Laplace Transform of Cosine",
"Laplace Transform of Sine",
"Definition:Laplace Transform",
"Injectivity of Laplace Transform"
] |
proofwiki-14836 | Euler's Formula/Real Domain | Let $\theta \in \R$ be a real number.
Then:
:$e^{i \theta} = \cos \theta + i \sin \theta$ | As Sine Function is Absolutely Convergent and Cosine Function is Absolutely Convergent, we have:
{{begin-eqn}}
{{eqn | l = \cos \theta + i \sin \theta
| r = \sum_{n \mathop = 0}^\infty \paren {-1}^n \dfrac {\theta^{2 n} } {\paren {2 n}!} + i \sum_{n \mathop = 0}^\infty \paren {-1}^n \dfrac {\theta^{2 n + 1} } {\p... | Let $\theta \in \R$ be a [[Definition:Real Number|real number]].
Then:
:$e^{i \theta} = \cos \theta + i \sin \theta$ | As [[Sine Function is Absolutely Convergent]] and [[Cosine Function is Absolutely Convergent]], we have:
{{begin-eqn}}
{{eqn | l = \cos \theta + i \sin \theta
| r = \sum_{n \mathop = 0}^\infty \paren {-1}^n \dfrac {\theta^{2 n} } {\paren {2 n}!} + i \sum_{n \mathop = 0}^\infty \paren {-1}^n \dfrac {\theta^{2 n +... | Euler's Formula/Real Domain/Proof 5 | https://proofwiki.org/wiki/Euler's_Formula/Real_Domain | https://proofwiki.org/wiki/Euler's_Formula/Real_Domain/Proof_5 | [
"Euler's Formula",
"Exponential Function",
"Trigonometric Functions"
] | [
"Definition:Real Number"
] | [
"Sine Function is Absolutely Convergent",
"Cosine Function is Absolutely Convergent",
"Sum of Absolutely Convergent Series"
] |
proofwiki-14837 | Euler's Cosine Identity/Real Domain | :$\cos x = \dfrac {e^{i x} + e^{-i x} } 2$ | Recall the definition of the real cosine function:
{{begin-eqn}}
{{eqn | l = \cos x
| r = \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {x^{2 n} } {\paren {2 n!} }
| c =
}}
{{eqn | r = 1 - \frac {x^2} {2!} + \frac {x^4} {4!} - \frac {x^6} {6!} + \cdots + \paren {-1}^n \frac {x^{2 n} } {\paren {2 n}!} + \... | :$\cos x = \dfrac {e^{i x} + e^{-i x} } 2$ | Recall the definition of the [[Definition:Real Cosine Function|real cosine function]]:
{{begin-eqn}}
{{eqn | l = \cos x
| r = \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {x^{2 n} } {\paren {2 n!} }
| c =
}}
{{eqn | r = 1 - \frac {x^2} {2!} + \frac {x^4} {4!} - \frac {x^6} {6!} + \cdots + \paren {-1}^n... | Euler's Cosine Identity/Real Domain/Proof 1 | https://proofwiki.org/wiki/Euler's_Cosine_Identity/Real_Domain | https://proofwiki.org/wiki/Euler's_Cosine_Identity/Real_Domain/Proof_1 | [
"Euler's Cosine Identity",
"Cosine Function"
] | [] | [
"Definition:Cosine/Real Function",
"Definition:Exponential Function/Real/Power Series Expansion",
"Cosine Function is Absolutely Convergent",
"Definition:Even Integer",
"Definition:Odd Integer"
] |
proofwiki-14838 | Euler's Cosine Identity/Real Domain | :$\cos x = \dfrac {e^{i x} + e^{-i x} } 2$ | Recall Euler's Formula:
:$e^{i x} = \cos x + i \sin x$
Then, starting from the {{RHS}}:
{{begin-eqn}}
{{eqn | l = \frac {e^{i x} + e^{-i x} } 2
| r = \frac {\cos x + i \sin x + \map \cos {-x} + i \map \sin {-x} } 2
}}
{{eqn | r = \frac {\cos x + \map \cos {-x} } 2
| c = Sine Function is Odd
}}
{{eqn | r = \... | :$\cos x = \dfrac {e^{i x} + e^{-i x} } 2$ | Recall [[Euler's Formula/Real Domain|Euler's Formula]]:
:$e^{i x} = \cos x + i \sin x$
Then, starting from the {{RHS}}:
{{begin-eqn}}
{{eqn | l = \frac {e^{i x} + e^{-i x} } 2
| r = \frac {\cos x + i \sin x + \map \cos {-x} + i \map \sin {-x} } 2
}}
{{eqn | r = \frac {\cos x + \map \cos {-x} } 2
| c = [[... | Euler's Cosine Identity/Real Domain/Proof 2 | https://proofwiki.org/wiki/Euler's_Cosine_Identity/Real_Domain | https://proofwiki.org/wiki/Euler's_Cosine_Identity/Real_Domain/Proof_2 | [
"Euler's Cosine Identity",
"Cosine Function"
] | [] | [
"Euler's Formula/Real Domain",
"Sine Function is Odd",
"Cosine Function is Even"
] |
proofwiki-14839 | Euler's Cosine Identity/Real Domain | :$\cos x = \dfrac {e^{i x} + e^{-i x} } 2$ | {{begin-eqn}}
{{eqn | n = 1
| l = e^{i x}
| r = \cos x + i \sin x
| c = Euler's Formula
}}
{{eqn | n = 2
| l = e^{-i x}
| r = \cos x - i \sin x
| c = Euler's Formula: Corollary
}}
{{eqn | ll= \leadsto
| l = e^{i x} + e^{-i x}
| r = \paren {\cos x + i \sin x} + \paren {\co... | :$\cos x = \dfrac {e^{i x} + e^{-i x} } 2$ | {{begin-eqn}}
{{eqn | n = 1
| l = e^{i x}
| r = \cos x + i \sin x
| c = [[Euler's Formula/Real Domain|Euler's Formula]]
}}
{{eqn | n = 2
| l = e^{-i x}
| r = \cos x - i \sin x
| c = [[Euler's Formula/Real Domain/Corollary|Euler's Formula: Corollary]]
}}
{{eqn | ll= \leadsto
| l... | Euler's Cosine Identity/Real Domain/Proof 3 | https://proofwiki.org/wiki/Euler's_Cosine_Identity/Real_Domain | https://proofwiki.org/wiki/Euler's_Cosine_Identity/Real_Domain/Proof_3 | [
"Euler's Cosine Identity",
"Cosine Function"
] | [] | [
"Euler's Formula/Real Domain",
"Euler's Formula/Real Domain/Corollary"
] |
proofwiki-14840 | Euler's Cosine Identity/Real Domain | :$\cos x = \dfrac {e^{i x} + e^{-i x} } 2$ | Consider the differential equation:
:$(1): \quad D^2_x \map f x = -\map f x$
subject to the initial conditions:
:$(2): \quad \map f 0 = 1$
:$(3): \quad D_x \map f 0 = 0$
=== Step 1 ===
We will prove that $y = \cos x$ is a particular solution of $(1)$.
{{begin-eqn}}
{{eqn | l = y
| r = \cos x
| c =
}}
{{eqn... | :$\cos x = \dfrac {e^{i x} + e^{-i x} } 2$ | Consider the [[Definition:Second Order Ordinary Differential Equation|differential equation]]:
:$(1): \quad D^2_x \map f x = -\map f x$
subject to the [[Definition:Initial Condition|initial conditions]]:
:$(2): \quad \map f 0 = 1$
:$(3): \quad D_x \map f 0 = 0$
=== Step 1 ===
We will prove that $y = \cos x$ is a [... | Euler's Cosine Identity/Real Domain/Proof 4 | https://proofwiki.org/wiki/Euler's_Cosine_Identity/Real_Domain | https://proofwiki.org/wiki/Euler's_Cosine_Identity/Real_Domain/Proof_4 | [
"Euler's Cosine Identity",
"Cosine Function"
] | [] | [
"Definition:Second Order Ordinary Differential Equation",
"Definition:Initial Condition",
"Definition:Differential Equation/Solution/Particular Solution",
"Definition:Derivative/Higher Derivatives/Second Derivative",
"Derivative of Cosine Function",
"Derivative of Constant Multiple",
"Derivative of Sine... |
proofwiki-14841 | Euler's Sine Identity/Real Domain | :$\sin x = \dfrac {e^{i x} - e^{-i x} } {2 i}$ | Recall the definition of the sine function:
{{begin-eqn}}
{{eqn | l = \sin x
| r = \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {x^{2 n + 1} } {\paren {2 n + 1}!}
| c =
}}
{{eqn | r = x - \frac {x^3} {3!} + \frac {x^5} {5!} - \frac {x^7} {7!} + \cdots + \paren {-1}^n \frac {x^{2 n + 1} } {\paren {2 n + ... | :$\sin x = \dfrac {e^{i x} - e^{-i x} } {2 i}$ | Recall the definition of the [[Definition:Real Sine Function|sine function]]:
{{begin-eqn}}
{{eqn | l = \sin x
| r = \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {x^{2 n + 1} } {\paren {2 n + 1}!}
| c =
}}
{{eqn | r = x - \frac {x^3} {3!} + \frac {x^5} {5!} - \frac {x^7} {7!} + \cdots + \paren {-1}^n \... | Euler's Sine Identity/Real Domain/Proof 1 | https://proofwiki.org/wiki/Euler's_Sine_Identity/Real_Domain | https://proofwiki.org/wiki/Euler's_Sine_Identity/Real_Domain/Proof_1 | [
"Euler's Sine Identity",
"Sine Function"
] | [] | [
"Definition:Sine/Real Function",
"Definition:Exponential Function/Real/Power Series Expansion",
"Definition:Even Integer",
"Definition:Odd Integer"
] |
proofwiki-14842 | Euler's Sine Identity/Real Domain | :$\sin x = \dfrac {e^{i x} - e^{-i x} } {2 i}$ | Recall Euler's Formula:
:$e^{i x} = \cos x + i \sin x$
Then, starting from the {{RHS}}:
{{begin-eqn}}
{{eqn | l = \frac {e^{i x} - e^{-i x} }{2 i}
| r = \frac {\paren {\cos x + i \sin x} - \paren {\map \cos {-x} + i \map \sin {-x} } } {2 i}
}}
{{eqn | r = \frac {\paren {\cos x + i \sin x - \cos x - i \map \sin {-... | :$\sin x = \dfrac {e^{i x} - e^{-i x} } {2 i}$ | Recall [[Euler's Formula/Real Domain|Euler's Formula]]:
:$e^{i x} = \cos x + i \sin x$
Then, starting from the {{RHS}}:
{{begin-eqn}}
{{eqn | l = \frac {e^{i x} - e^{-i x} }{2 i}
| r = \frac {\paren {\cos x + i \sin x} - \paren {\map \cos {-x} + i \map \sin {-x} } } {2 i}
}}
{{eqn | r = \frac {\paren {\cos x ... | Euler's Sine Identity/Real Domain/Proof 2 | https://proofwiki.org/wiki/Euler's_Sine_Identity/Real_Domain | https://proofwiki.org/wiki/Euler's_Sine_Identity/Real_Domain/Proof_2 | [
"Euler's Sine Identity",
"Sine Function"
] | [] | [
"Euler's Formula/Real Domain",
"Cosine Function is Even",
"Sine Function is Odd"
] |
proofwiki-14843 | Euler's Sine Identity/Real Domain | :$\sin x = \dfrac {e^{i x} - e^{-i x} } {2 i}$ | {{begin-eqn}}
{{eqn | n = 1
| l = e^{i x}
| r = \cos x + i \sin x
| c = Euler's Formula
}}
{{eqn | n = 2
| l = e^{-i x}
| r = \cos x - i \sin x
| c = Euler's Formula: Corollary
}}
{{eqn | ll= \leadsto
| l = e^{i x} - e^{-i x}
| r = \paren {\cos x + i \sin x} - \paren {\co... | :$\sin x = \dfrac {e^{i x} - e^{-i x} } {2 i}$ | {{begin-eqn}}
{{eqn | n = 1
| l = e^{i x}
| r = \cos x + i \sin x
| c = [[Euler's Formula/Real Domain|Euler's Formula]]
}}
{{eqn | n = 2
| l = e^{-i x}
| r = \cos x - i \sin x
| c = [[Euler's Formula/Real Domain/Corollary|Euler's Formula: Corollary]]
}}
{{eqn | ll= \leadsto
| l... | Euler's Sine Identity/Real Domain/Proof 3 | https://proofwiki.org/wiki/Euler's_Sine_Identity/Real_Domain | https://proofwiki.org/wiki/Euler's_Sine_Identity/Real_Domain/Proof_3 | [
"Euler's Sine Identity",
"Sine Function"
] | [] | [
"Euler's Formula/Real Domain",
"Euler's Formula/Real Domain/Corollary"
] |
proofwiki-14844 | Reverse Triangle Inequality/Normed Division Ring | Let $\struct {R, \norm {\,\cdot\,} }$ be a normed division ring.
Then:
:$\forall x, y \in R: \norm {x - y} \ge \bigsize {\norm x - \norm y}$ | Let $0$ be the zero of $\struct {R, \norm {\,\cdot\,} }$.
Let $d$ denote the metric induced by $\norm {\, \cdot \,}$, that is:
:$\map d {x, y} = \norm {x - y}$
From Metric Induced by Norm on Normed Division Ring is Metric we have that $d$ is indeed a metric.
Then, from the Reverse Triangle Inequality as applied to metr... | Let $\struct {R, \norm {\,\cdot\,} }$ be a [[Definition:Normed Division Ring|normed division ring]].
Then:
:$\forall x, y \in R: \norm {x - y} \ge \bigsize {\norm x - \norm y}$ | Let $0$ be the [[Definition:Ring Zero|zero]] of $\struct {R, \norm {\,\cdot\,} }$.
Let $d$ denote the [[Definition:Metric Induced by Norm|metric induced by $\norm {\, \cdot \,}$]], that is:
:$\map d {x, y} = \norm {x - y}$
From [[Metric Induced by Norm on Normed Division Ring is Metric]] we have that $d$ is indeed a ... | Reverse Triangle Inequality/Normed Division Ring | https://proofwiki.org/wiki/Reverse_Triangle_Inequality/Normed_Division_Ring | https://proofwiki.org/wiki/Reverse_Triangle_Inequality/Normed_Division_Ring | [
"Triangle Inequality",
"Normed Division Rings"
] | [
"Definition:Normed Division Ring"
] | [
"Definition:Ring Zero",
"Definition:Metric Induced by Norm",
"Metric Induced by Norm on Normed Division Ring is Metric",
"Definition:Metric Space/Metric",
"Reverse Triangle Inequality",
"Definition:Metric Space",
"Category:Triangle Inequality",
"Category:Normed Division Rings"
] |
proofwiki-14845 | Sequence Converges to Within Half Limit/Normed Division Ring | Let $\struct {R, \norm {\, \cdot \,} }$ be a normed division ring with zero $0$.
Let $\sequence {x_n}$ be a sequence in $R$.
Let $\sequence {x_n}$ be convergent in the norm $\norm {\, \cdot \,}$ to the following limit:
:$\ds \lim_{n \mathop \to \infty} x_n = l \ne 0$
Then:
:$\exists N: \forall n > N: \norm {x_n} > \dfr... | Since $l \ne 0$, by {{Norm-axiom-mult|1}}:
:$\norm l > 0$
Let us choose $N$ such that:
:$\forall n > N: \norm {x_n - l} < \dfrac {\norm l} 2$
Then:
{{begin-eqn}}
{{eqn | l = \norm {x_n - l}
| o = <
| r = \frac {\norm l} 2
| c =
}}
{{eqn | ll= \leadsto
| l = \norm l - \norm {x_n}
| o = \le... | Let $\struct {R, \norm {\, \cdot \,} }$ be a [[Definition:Normed Division Ring|normed division ring]] with [[Definition:Ring Zero|zero]] $0$.
Let $\sequence {x_n}$ be a [[Definition:Sequence|sequence in $R$]].
Let $\sequence {x_n}$ be [[Definition:Convergent Sequence in Normed Division Ring|convergent in the norm]] $... | Since $l \ne 0$, by {{Norm-axiom-mult|1}}:
:$\norm l > 0$
Let us choose $N$ such that:
:$\forall n > N: \norm {x_n - l} < \dfrac {\norm l} 2$
Then:
{{begin-eqn}}
{{eqn | l = \norm {x_n - l}
| o = <
| r = \frac {\norm l} 2
| c =
}}
{{eqn | ll= \leadsto
| l = \norm l - \norm {x_n}
| o =... | Sequence Converges to Within Half Limit/Normed Division Ring | https://proofwiki.org/wiki/Sequence_Converges_to_Within_Half_Limit/Normed_Division_Ring | https://proofwiki.org/wiki/Sequence_Converges_to_Within_Half_Limit/Normed_Division_Ring | [
"Sequences",
"Limits of Sequences",
"Normed Division Rings"
] | [
"Definition:Normed Division Ring",
"Definition:Ring Zero",
"Definition:Sequence",
"Definition:Convergent Sequence/Normed Division Ring",
"Definition:Limit of Sequence/Normed Division Ring"
] | [
"Reverse Triangle Inequality",
"Category:Sequences",
"Category:Limits of Sequences",
"Category:Normed Division Rings"
] |
proofwiki-14846 | Limit of Subsequence equals Limit of Sequence/Normed Division Ring | Let $\struct {R, \norm {\, \cdot \,} }$ be a normed division ring with zero: $0$.
Let $\sequence {x_n}$ be a sequence in $R$.
Let $\sequence {x_n}$ be convergent in the norm $\norm {\, \cdot \,}$ to the following limit:
:$\ds \lim_{n \mathop \to \infty} x_n = l$
Let $\sequence {x_{n_r} }$ be a subsequence of $\sequence... | Let $d$ denote the metric induced by $\norm {\, \cdot \,}$, that is,
:$d \tuple {x, y} = \norm {x - y}$
By definition of convergence in a normed division ring:
:$\sequence {x_n}$ converges to $l$ in $\struct {R, \norm {\, \cdot \,} }$ {{iff}} $\sequence {x_n}$ converges to $l$ in the metric space $\struct {R, d}$.
We c... | Let $\struct {R, \norm {\, \cdot \,} }$ be a [[Definition:Normed Division Ring|normed division ring]] with [[Definition:Ring Zero|zero]]: $0$.
Let $\sequence {x_n}$ be a [[Definition:Sequence|sequence in $R$]].
Let $\sequence {x_n}$ be [[Definition:Convergent Sequence in Normed Division Ring|convergent in the norm]] ... | Let $d$ denote the [[Definition:Metric Induced by Norm|metric induced by $\norm {\, \cdot \,}$]], that is,
:$d \tuple {x, y} = \norm {x - y}$
By definition of [[Definition:Convergent Sequence in Normed Division Ring|convergence in a normed division ring]]:
:$\sequence {x_n}$ converges to $l$ in $\struct {R, \norm {\,... | Limit of Subsequence equals Limit of Sequence/Normed Division Ring | https://proofwiki.org/wiki/Limit_of_Subsequence_equals_Limit_of_Sequence/Normed_Division_Ring | https://proofwiki.org/wiki/Limit_of_Subsequence_equals_Limit_of_Sequence/Normed_Division_Ring | [
"Sequences",
"Normed Division Rings"
] | [
"Definition:Normed Division Ring",
"Definition:Ring Zero",
"Definition:Sequence",
"Definition:Convergent Sequence/Normed Division Ring",
"Definition:Limit of Sequence/Normed Division Ring",
"Definition:Subsequence",
"Definition:Convergent Sequence/Normed Division Ring"
] | [
"Definition:Metric Induced by Norm",
"Definition:Convergent Sequence/Normed Division Ring",
"Definition:Convergent Sequence/Metric Space",
"Limit of Subsequence equals Limit of Sequence/Metric Space",
"Definition:Metric Space",
"Definition:Normed Division Ring",
"Category:Sequences",
"Category:Normed ... |
proofwiki-14847 | Combination Theorem for Sequences/Normed Division Ring/Quotient Rule | Suppose $m \ne 0$.
Then:
:$\exists k \in \N : \forall n \in \N: y_{k + n} \ne 0$
and the sequences:
:$\sequence {x_{k + n} \ {y_{k + n} }^{-1} }$ and $\sequence { {y_{k + n} }^{-1} \ x_{k + n} }$ are well-defined and convergent with:
:$\ds \lim_{n \mathop \to \infty} x_{k + n} \ {y_{k + n} }^{-1} = l m^{-1}$
:$\ds \lim... | From Inverse Rule for Sequences in Normed Division Ring:
:$\exists k \in \N : \forall n \in \N : y_{k + n} \ne 0$
and the sequence:
:$\sequence { {y_{k + n} }^{-1} }$
is well-defined and convergent with:
:$\ds \lim_{n \mathop \to \infty} {y_{k + n} }^{-1} = m^{-1}$
From Limit of Subsequence equals Limit of Sequence, $\... | Suppose $m \ne 0$.
Then:
:$\exists k \in \N : \forall n \in \N: y_{k + n} \ne 0$
and the [[Definition:Sequence|sequences]]:
:$\sequence {x_{k + n} \ {y_{k + n} }^{-1} }$ and $\sequence { {y_{k + n} }^{-1} \ x_{k + n} }$ are well-defined and [[Definition:Convergent Sequence in Normed Division Ring|convergent]] with:
:... | From [[Inverse Rule for Sequences in Normed Division Ring]]:
:$\exists k \in \N : \forall n \in \N : y_{k + n} \ne 0$
and the [[Definition:Sequence|sequence]]:
:$\sequence { {y_{k + n} }^{-1} }$
is well-defined and [[Definition:Convergent Sequence in Normed Division Ring|convergent]] with:
:$\ds \lim_{n \mathop \to \i... | Combination Theorem for Sequences/Normed Division Ring/Quotient Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Sequences/Normed_Division_Ring/Quotient_Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Sequences/Normed_Division_Ring/Quotient_Rule | [
"Combination Theorem for Sequences in Normed Division Rings"
] | [
"Definition:Sequence",
"Definition:Convergent Sequence/Normed Division Ring"
] | [
"Combination Theorem for Sequences/Normed Division Ring/Inverse Rule",
"Definition:Sequence",
"Definition:Convergent Sequence/Normed Division Ring",
"Limit of Subsequence equals Limit of Sequence/Normed Division Ring",
"Definition:Convergent Sequence/Normed Division Ring",
"Combination Theorem for Sequenc... |
proofwiki-14848 | Real Sine Function is Bounded | :$\size {\sin x} \le 1$ | From the algebraic definition of the real sine function:
:$\ds \sin x = \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {x^{2 n + 1} } {\paren {2 n + 1}!}$
it follows that $\sin x$ is a real function.
Thus $\sin^2 x \ge 0$.
From Sum of Squares of Sine and Cosine, we have that $\cos^2 x + \sin^2 x = 1$.
Thus it follows ... | :$\size {\sin x} \le 1$ | From the algebraic definition of the [[Definition:Real Sine Function|real sine function]]:
:$\ds \sin x = \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {x^{2 n + 1} } {\paren {2 n + 1}!}$
it follows that $\sin x$ is a [[Definition:Real Function|real function]].
Thus $\sin^2 x \ge 0$.
From [[Sum of Squares of Sine... | Real Sine Function is Bounded | https://proofwiki.org/wiki/Real_Sine_Function_is_Bounded | https://proofwiki.org/wiki/Real_Sine_Function_is_Bounded | [
"Sine Function"
] | [] | [
"Definition:Sine/Real Function",
"Definition:Real Function",
"Sum of Squares of Sine and Cosine",
"Ordering of Squares in Reals",
"Definition:Absolute Value"
] |
proofwiki-14849 | Complex Cosine Function is Unbounded | The complex cosine function is unbounded. | Let $K \in \R_{>0}$ be an arbitrary real number.
Let $p = \ln {2 K}$.
Let $z = i p$, where $i$ denotes the imaginary unit.
Then:
{{begin-eqn}}
{{eqn | l = \cos z
| r = \dfrac {\map \exp {i \paren {i p} } + \map \exp {-i \paren {i p} } } 2
| c = Euler's Cosine Identity
}}
{{eqn | r = \dfrac {\exp p + \map \e... | The [[Definition:Complex Cosine Function|complex cosine function]] is [[Definition:Unbounded Complex-Valued Function|unbounded]]. | Let $K \in \R_{>0}$ be an arbitrary [[Definition:Real Number|real number]].
Let $p = \ln {2 K}$.
Let $z = i p$, where $i$ denotes the [[Definition:Imaginary Unit|imaginary unit]].
Then:
{{begin-eqn}}
{{eqn | l = \cos z
| r = \dfrac {\map \exp {i \paren {i p} } + \map \exp {-i \paren {i p} } } 2
| c = [... | Complex Cosine Function is Unbounded/Proof 1 | https://proofwiki.org/wiki/Complex_Cosine_Function_is_Unbounded | https://proofwiki.org/wiki/Complex_Cosine_Function_is_Unbounded/Proof_1 | [
"Cosine Function",
"Complex Cosine Function is Unbounded"
] | [
"Definition:Cosine/Complex Function",
"Definition:Bounded Mapping/Complex-Valued/Unbounded"
] | [
"Definition:Real Number",
"Definition:Complex Number/Imaginary Unit",
"Euler's Cosine Identity",
"Definition:Bounded Mapping/Complex-Valued/Unbounded"
] |
proofwiki-14850 | Complex Cosine Function is Unbounded | The complex cosine function is unbounded. | By Complex Cosine Function is Entire, we have that $\cos$ is an entire function.
{{AimForCont}} that $\cos$ is a bounded function.
By Liouville's Theorem, we have that $\cos$ is a constant function.
However, by Cosine of Zero is One:
:$\cos 0 = 1$
and by Cosine of Right Angle:
:$\cos \dfrac \pi 2 = 0$
Therefore, $\c... | The [[Definition:Complex Cosine Function|complex cosine function]] is [[Definition:Unbounded Complex-Valued Function|unbounded]]. | By [[Complex Cosine Function is Entire]], we have that $\cos$ is an [[Definition:Entire Function|entire function]].
{{AimForCont}} that $\cos$ is a [[Definition:Bounded Complex-Valued Function|bounded function]].
By [[Liouville's Theorem (Complex Analysis)|Liouville's Theorem]], we have that $\cos$ is a [[Definition... | Complex Cosine Function is Unbounded/Proof 2 | https://proofwiki.org/wiki/Complex_Cosine_Function_is_Unbounded | https://proofwiki.org/wiki/Complex_Cosine_Function_is_Unbounded/Proof_2 | [
"Cosine Function",
"Complex Cosine Function is Unbounded"
] | [
"Definition:Cosine/Complex Function",
"Definition:Bounded Mapping/Complex-Valued/Unbounded"
] | [
"Complex Cosine Function is Entire",
"Definition:Entire Function",
"Definition:Bounded Mapping/Complex-Valued",
"Liouville's Theorem (Complex Analysis)",
"Definition:Constant Mapping",
"Cosine of Zero is One",
"Cosine of Right Angle",
"Definition:Constant Mapping",
"Definition:Contradiction",
"Def... |
proofwiki-14851 | Real Part of Complex Exponential Function | Let $z = x + i y \in \C$ be a complex number, where $x, y \in \R$.
Let $\exp z$ denote the complex exponential function.
Then:
:$\map \Re {\exp z} = e^x \cos y$
where:
:$\Re z$ denotes the real part of a complex number $z$
:$e^x$ denotes the real exponential function of $x$
:$\cos y$ denotes the real cosine function of... | From the definition of the complex exponential function:
:$\exp z := e^x \paren {\cos y + i \sin y}$
The result follows by definition of the real part of a complex number.
{{qed}} | Let $z = x + i y \in \C$ be a [[Definition:Complex Number|complex number]], where $x, y \in \R$.
Let $\exp z$ denote the [[Definition:Complex Exponential Function|complex exponential function]].
Then:
:$\map \Re {\exp z} = e^x \cos y$
where:
:$\Re z$ denotes the [[Definition:Real Part|real part]] of a [[Definition:Co... | From the definition of the [[Definition:Exponential Function/Complex/Real Functions|complex exponential function]]:
:$\exp z := e^x \paren {\cos y + i \sin y}$
The result follows by definition of the [[Definition:Real Part|real part]] of a [[Definition:Complex Number|complex number]].
{{qed}} | Real Part of Complex Exponential Function | https://proofwiki.org/wiki/Real_Part_of_Complex_Exponential_Function | https://proofwiki.org/wiki/Real_Part_of_Complex_Exponential_Function | [
"Exponential Function"
] | [
"Definition:Complex Number",
"Definition:Exponential Function/Complex",
"Definition:Complex Number/Real Part",
"Definition:Complex Number",
"Definition:Exponential Function/Real",
"Definition:Cosine/Real Function"
] | [
"Definition:Exponential Function/Complex/Real Functions",
"Definition:Complex Number/Real Part",
"Definition:Complex Number"
] |
proofwiki-14852 | Imaginary Part of Complex Exponential Function | Let $z = x + i y \in \C$ be a complex number, where $x, y \in \R$.
Let $\exp z$ denote the complex exponential function.
Then:
:$\map \Im {\exp z} = e^x \sin y$
where:
:$\map \Im z$ denotes the imaginary part of a complex number $z$
:$e^x$ denotes the real exponential function of $x$
:$\sin y$ denotes the real sine fun... | From the definition of the complex exponential function:
:$\exp z := e^x \paren {\cos y + i \sin y}$
The result follows by definition of the imaginary part of a complex number.
{{qed}} | Let $z = x + i y \in \C$ be a [[Definition:Complex Number|complex number]], where $x, y \in \R$.
Let $\exp z$ denote the [[Definition:Complex Exponential Function|complex exponential function]].
Then:
:$\map \Im {\exp z} = e^x \sin y$
where:
:$\map \Im z$ denotes the [[Definition:Imaginary Part|imaginary part]] of a ... | From the definition of the [[Definition:Exponential Function/Complex/Real Functions|complex exponential function]]:
:$\exp z := e^x \paren {\cos y + i \sin y}$
The result follows by definition of the [[Definition:Imaginary Part|imaginary part]] of a [[Definition:Complex Number|complex number]].
{{qed}} | Imaginary Part of Complex Exponential Function | https://proofwiki.org/wiki/Imaginary_Part_of_Complex_Exponential_Function | https://proofwiki.org/wiki/Imaginary_Part_of_Complex_Exponential_Function | [
"Exponential Function"
] | [
"Definition:Complex Number",
"Definition:Exponential Function/Complex",
"Definition:Complex Number/Imaginary Part",
"Definition:Complex Number",
"Definition:Exponential Function/Real",
"Definition:Sine/Real Function"
] | [
"Definition:Exponential Function/Complex/Real Functions",
"Definition:Complex Number/Imaginary Part",
"Definition:Complex Number"
] |
proofwiki-14853 | Argument of Exponential is Imaginary Part plus Multiple of 2 Pi | Let $z \in \C$ be a complex number.
Let $\exp z$ denote the complex exponential of $z$.
Let $\arg z$ denote the argument of $z$.
Then:
:$\map \arg {\exp z} = \set {\Im z + 2 k \pi: k \in \Z}$
where $\Im z$ denotes the imaginary part of $z$. | Let $z = x + i y$.
Let $\theta \in \map \arg {\exp z}$.
We have:
{{begin-eqn}}
{{eqn | l = \exp z
| r = e^x \paren {\cos y + i \sin y}
| c = {{Defof|Exponential Function/Complex|Exponential Function|subdef = Real Functions}}
}}
{{eqn | ll= \leadsto
| l = y
| o = \in
| r = \map \arg {\exp z... | Let $z \in \C$ be a [[Definition:Complex Number|complex number]].
Let $\exp z$ denote the [[Definition:Complex Exponential Function|complex exponential]] of $z$.
Let $\arg z$ denote the [[Definition:Argument of Complex Number|argument]] of $z$.
Then:
:$\map \arg {\exp z} = \set {\Im z + 2 k \pi: k \in \Z}$
where ... | Let $z = x + i y$.
Let $\theta \in \map \arg {\exp z}$.
We have:
{{begin-eqn}}
{{eqn | l = \exp z
| r = e^x \paren {\cos y + i \sin y}
| c = {{Defof|Exponential Function/Complex|Exponential Function|subdef = Real Functions}}
}}
{{eqn | ll= \leadsto
| l = y
| o = \in
| r = \map \arg {\ex... | Argument of Exponential is Imaginary Part plus Multiple of 2 Pi | https://proofwiki.org/wiki/Argument_of_Exponential_is_Imaginary_Part_plus_Multiple_of_2_Pi | https://proofwiki.org/wiki/Argument_of_Exponential_is_Imaginary_Part_plus_Multiple_of_2_Pi | [
"Argument of Complex Number",
"Exponential Function"
] | [
"Definition:Complex Number",
"Definition:Exponential Function/Complex",
"Definition:Argument of Complex Number",
"Definition:Complex Number/Imaginary Part"
] | [] |
proofwiki-14854 | Real Part of Sine of Complex Number | Let $z = x + i y \in \C$ be a complex number, where $x, y \in \R$.
Let $\sin z$ denote the complex sine function.
Then:
:$\map \Re {\sin z} = \sin x \cosh y$
where:
:$\map \Re z$ denotes the real part of a complex number $z$
:$\sin$ denotes the sine function (real and complex)
:$\cosh$ denotes the hyperbolic cosine fun... | From Sine of Complex Number:
:$\map \sin {x + i y} = \sin x \cosh y + i \cos x \sinh y$
The result follows by definition of the real part of a complex number.
{{qed}} | Let $z = x + i y \in \C$ be a [[Definition:Complex Number|complex number]], where $x, y \in \R$.
Let $\sin z$ denote the [[Definition:Complex Sine Function|complex sine function]].
Then:
:$\map \Re {\sin z} = \sin x \cosh y$
where:
:$\map \Re z$ denotes the [[Definition:Real Part|real part]] of a [[Definition:Complex... | From [[Sine of Complex Number]]:
:$\map \sin {x + i y} = \sin x \cosh y + i \cos x \sinh y$
The result follows by definition of the [[Definition:Real Part|real part]] of a [[Definition:Complex Number|complex number]].
{{qed}} | Real Part of Sine of Complex Number | https://proofwiki.org/wiki/Real_Part_of_Sine_of_Complex_Number | https://proofwiki.org/wiki/Real_Part_of_Sine_of_Complex_Number | [
"Sine Function"
] | [
"Definition:Complex Number",
"Definition:Sine/Complex Function",
"Definition:Complex Number/Real Part",
"Definition:Complex Number",
"Definition:Sine",
"Definition:Sine/Real Function",
"Definition:Sine/Complex Function",
"Definition:Hyperbolic Cosine"
] | [
"Sine of Complex Number",
"Definition:Complex Number/Real Part",
"Definition:Complex Number"
] |
proofwiki-14855 | Imaginary Part of Sine of Complex Number | Let $z = x + i y \in \C$ be a complex number, where $x, y \in \R$.
Let $\sin z$ denote the complex sine function.
Then:
:$\Im \paren {\sin z} = \cos x \sinh y$
where:
:$\Im z$ denotes the imaginary part of a complex number $z$
:$\sin$ denotes the complex sine function
:$\cos$ denotes the real cosine function
:$\sinh$ d... | From Sine of Complex Number:
:$\sin \paren {x + i y} = \sin x \cosh y + i \cos x \sinh y$
The result follows by definition of the imaginary part of a complex number.
{{qed}} | Let $z = x + i y \in \C$ be a [[Definition:Complex Number|complex number]], where $x, y \in \R$.
Let $\sin z$ denote the [[Definition:Complex Sine Function|complex sine function]].
Then:
:$\Im \paren {\sin z} = \cos x \sinh y$
where:
:$\Im z$ denotes the [[Definition:Imaginary Part|imaginary part]] of a [[Definition:... | From [[Sine of Complex Number]]:
:$\sin \paren {x + i y} = \sin x \cosh y + i \cos x \sinh y$
The result follows by definition of the [[Definition:Imaginary Part|imaginary part]] of a [[Definition:Complex Number|complex number]].
{{qed}} | Imaginary Part of Sine of Complex Number | https://proofwiki.org/wiki/Imaginary_Part_of_Sine_of_Complex_Number | https://proofwiki.org/wiki/Imaginary_Part_of_Sine_of_Complex_Number | [
"Sine Function"
] | [
"Definition:Complex Number",
"Definition:Sine/Complex Function",
"Definition:Complex Number/Imaginary Part",
"Definition:Complex Number",
"Definition:Sine/Complex Function",
"Definition:Cosine/Real Function",
"Definition:Hyperbolic Sine"
] | [
"Sine of Complex Number",
"Definition:Complex Number/Imaginary Part",
"Definition:Complex Number"
] |
proofwiki-14856 | Modulus of Sine of Complex Number | Let $z = x + i y \in \C$ be a complex number, where $x, y \in \R$.
Let $\sin z$ denote the complex sine function.
Then:
:$\cmod {\sin z} = \sqrt {\sin^2 x + \sinh^2 y}$
where:
:$\cmod z$ denotes the modulus of a complex number $z$
:$\sin x$ denotes the real sine function
:$\sinh$ denotes the hyperbolic sine function. | {{begin-eqn}}
{{eqn | l = \sin \paren {x + i y}
| r = \sin x \cosh y + i \cos x \sinh y
| c = Sine of Complex Number
}}
{{eqn | ll= \leadsto
| l = \cmod {\sin z}^2
| r = \paren {\sin x \cosh y}^2 + \paren {\cos x \sinh y}^2
| c = {{Defof|Complex Modulus}}
}}
{{eqn | r = \sin^2 x \paren {1 ... | Let $z = x + i y \in \C$ be a [[Definition:Complex Number|complex number]], where $x, y \in \R$.
Let $\sin z$ denote the [[Definition:Complex Sine Function|complex sine function]].
Then:
:$\cmod {\sin z} = \sqrt {\sin^2 x + \sinh^2 y}$
where:
:$\cmod z$ denotes the [[Definition:Complex Modulus|modulus]] of a [[Defini... | {{begin-eqn}}
{{eqn | l = \sin \paren {x + i y}
| r = \sin x \cosh y + i \cos x \sinh y
| c = [[Sine of Complex Number]]
}}
{{eqn | ll= \leadsto
| l = \cmod {\sin z}^2
| r = \paren {\sin x \cosh y}^2 + \paren {\cos x \sinh y}^2
| c = {{Defof|Complex Modulus}}
}}
{{eqn | r = \sin^2 x \paren... | Modulus of Sine of Complex Number | https://proofwiki.org/wiki/Modulus_of_Sine_of_Complex_Number | https://proofwiki.org/wiki/Modulus_of_Sine_of_Complex_Number | [
"Sine Function"
] | [
"Definition:Complex Number",
"Definition:Sine/Complex Function",
"Definition:Complex Modulus",
"Definition:Complex Number",
"Definition:Sine/Real Function",
"Definition:Hyperbolic Sine"
] | [
"Sine of Complex Number",
"Difference of Squares of Hyperbolic Cosine and Sine",
"Sum of Squares of Sine and Cosine"
] |
proofwiki-14857 | Inverse Tangent of i | The inverse tangent of $i$ is not defined. | {{AimForCont}} $\tan z_0 = i$.
{{begin-eqn}}
{{eqn | l = \dfrac {\sin z_0} {\cos z_0}
| r = i
| c = {{Defof|Tangent Function}}
}}
{{eqn | ll= \leadsto
| l = \sin z_0
| r = i \cos z_0
| c =
}}
{{eqn | ll= \leadsto
| l = \sin^2 z_0
| r = -\cos^2 z_0
| c =
}}
{{eqn | ll= \... | The [[Definition:Complex Inverse Tangent|inverse tangent]] of $i$ is not defined. | {{AimForCont}} $\tan z_0 = i$.
{{begin-eqn}}
{{eqn | l = \dfrac {\sin z_0} {\cos z_0}
| r = i
| c = {{Defof|Tangent Function}}
}}
{{eqn | ll= \leadsto
| l = \sin z_0
| r = i \cos z_0
| c =
}}
{{eqn | ll= \leadsto
| l = \sin^2 z_0
| r = -\cos^2 z_0
| c =
}}
{{eqn | ll= ... | Inverse Tangent of i | https://proofwiki.org/wiki/Inverse_Tangent_of_i | https://proofwiki.org/wiki/Inverse_Tangent_of_i | [
"Inverse Tangent"
] | [
"Definition:Inverse Tangent/Complex"
] | [
"Definition:Contradiction",
"Sum of Squares of Sine and Cosine",
"Proof by Contradiction"
] |
proofwiki-14858 | Cauchy-Hadamard Theorem/Complex Case | Let $\xi \in \C$ be a complex number.
Let $\ds \map S z = \sum_{n \mathop = 0}^\infty a_n \paren {z - \xi}^n$ be a (complex) power series about $\xi$.
Then the radius of convergence $R$ of $\map S z$ is given by:
:$\ds \dfrac 1 R = \limsup_{n \mathop \to \infty} \cmod {a_n}^{1/n}$
If:
:$\ds \limsup_{n \mathop \to \inft... | Let $\epsilon \in \R_{>0}$, and let $z \in \C$.
Suppose that $\cmod {z - \xi} = R - \epsilon$.
By definition of radius of convergence, it follows that $S \paren z$ is absolutely convergent.
From the $n$th Root Test:
:$\ds \limsup_{n \mathop \to \infty} \cmod {a_n \paren {z - \xi}^n}^{1/n} \le 1$
By Multiple Rule for Co... | Let $\xi \in \C$ be a [[Definition:Complex Number|complex number]].
Let $\ds \map S z = \sum_{n \mathop = 0}^\infty a_n \paren {z - \xi}^n$ be a [[Definition:Complex Power Series|(complex) power series]] about $\xi$.
Then the [[Definition:Radius of Convergence of Complex Power Series|radius of convergence]] $R$ of $... | Let $\epsilon \in \R_{>0}$, and let $z \in \C$.
Suppose that $\cmod {z - \xi} = R - \epsilon$.
By definition of [[Definition:Radius of Convergence of Complex Power Series|radius of convergence]], it follows that $S \paren z$ is [[Definition:Absolutely Convergent Series|absolutely convergent]].
From the [[Nth Root Te... | Cauchy-Hadamard Theorem/Complex Case/Proof 1 | https://proofwiki.org/wiki/Cauchy-Hadamard_Theorem/Complex_Case | https://proofwiki.org/wiki/Cauchy-Hadamard_Theorem/Complex_Case/Proof_1 | [
"Cauchy-Hadamard Theorem",
"Complex Power Series"
] | [
"Definition:Complex Number",
"Definition:Power Series/Complex Domain",
"Definition:Radius of Convergence/Complex Domain",
"Definition:Radius of Convergence/Complex Domain",
"Definition:Infinite",
"Definition:Absolutely Convergent Series"
] | [
"Definition:Radius of Convergence/Complex Domain",
"Definition:Absolutely Convergent Series",
"Nth Root Test",
"Combination Theorem for Sequences/Complex/Multiple Rule",
"Definition:Divergent Series",
"Nth Root Test"
] |
proofwiki-14859 | Cauchy-Hadamard Theorem/Complex Case | Let $\xi \in \C$ be a complex number.
Let $\ds \map S z = \sum_{n \mathop = 0}^\infty a_n \paren {z - \xi}^n$ be a (complex) power series about $\xi$.
Then the radius of convergence $R$ of $\map S z$ is given by:
:$\ds \dfrac 1 R = \limsup_{n \mathop \to \infty} \cmod {a_n}^{1/n}$
If:
:$\ds \limsup_{n \mathop \to \inft... | Let $L = \limsup \cmod {a_n}^{1/n}$.
We will consider only the case $0 < L < \infty$, as the cases $L = 0$ and $L = \infty$ follow quite simply from this one.
We have that:
:$\forall r \in \closedint 0 {\dfrac 1 L}: L < \dfrac 1 r$
Thus there exists $\epsilon \in \R_{> 0}$ such that:
:$L + \epsilon < \dfrac 1 R$
and so... | Let $\xi \in \C$ be a [[Definition:Complex Number|complex number]].
Let $\ds \map S z = \sum_{n \mathop = 0}^\infty a_n \paren {z - \xi}^n$ be a [[Definition:Complex Power Series|(complex) power series]] about $\xi$.
Then the [[Definition:Radius of Convergence of Complex Power Series|radius of convergence]] $R$ of $... | Let $L = \limsup \cmod {a_n}^{1/n}$.
We will consider only the case $0 < L < \infty$, as the cases $L = 0$ and $L = \infty$ follow quite simply from this one.
We have that:
:$\forall r \in \closedint 0 {\dfrac 1 L}: L < \dfrac 1 r$
Thus there exists $\epsilon \in \R_{> 0}$ such that:
:$L + \epsilon < \dfrac 1 R$
and... | Cauchy-Hadamard Theorem/Complex Case/Proof 2 | https://proofwiki.org/wiki/Cauchy-Hadamard_Theorem/Complex_Case | https://proofwiki.org/wiki/Cauchy-Hadamard_Theorem/Complex_Case/Proof_2 | [
"Cauchy-Hadamard Theorem",
"Complex Power Series"
] | [
"Definition:Complex Number",
"Definition:Power Series/Complex Domain",
"Definition:Radius of Convergence/Complex Domain",
"Definition:Radius of Convergence/Complex Domain",
"Definition:Infinite",
"Definition:Absolutely Convergent Series"
] | [] |
proofwiki-14860 | Cauchy-Hadamard Theorem/Real Case | Let $\xi \in \R$ be a real number.
Let $\ds \map S x = \sum_{n \mathop = 0}^\infty a_n \paren {x - \xi}^n$ be a power series about $\xi$.
Then the radius of convergence $R$ of $S \paren x$ is given by:
:$\ds \frac 1 R = \limsup_{n \mathop \to \infty} \size {a_n}^{1/n}$
If:
:$\ds \frac 1 R = \limsup_{n \mathop \to \inft... | From the $n$th root test, $S \paren x$ is convergent if $\ds \limsup_{n \mathop \to \infty} \size {a_n \paren {x - \xi}^n}^{1/n} < 1$.
Thus:
{{begin-eqn}}
{{eqn | l = \size {a_n \paren {x - \xi}^n}^{1/n}
| o = <
| r = 1
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = \size {a_n}^{1/n} \size {x - \xi... | Let $\xi \in \R$ be a [[Definition:Real Number|real number]].
Let $\ds \map S x = \sum_{n \mathop = 0}^\infty a_n \paren {x - \xi}^n$ be a [[Definition:Power Series|power series]] about $\xi$.
Then the [[Definition:Radius of Convergence of Real Power Series|radius of convergence]] $R$ of $S \paren x$ is given by:
:$... | From the [[Nth Root Test|$n$th root test]], $S \paren x$ is [[Definition:Convergent Series|convergent]] if $\ds \limsup_{n \mathop \to \infty} \size {a_n \paren {x - \xi}^n}^{1/n} < 1$.
Thus:
{{begin-eqn}}
{{eqn | l = \size {a_n \paren {x - \xi}^n}^{1/n}
| o = <
| r = 1
| c =
}}
{{eqn | ll= \leadst... | Cauchy-Hadamard Theorem/Real Case | https://proofwiki.org/wiki/Cauchy-Hadamard_Theorem/Real_Case | https://proofwiki.org/wiki/Cauchy-Hadamard_Theorem/Real_Case | [
"Cauchy-Hadamard Theorem",
"Real Power Series"
] | [
"Definition:Real Number",
"Definition:Power Series",
"Definition:Radius of Convergence/Real Domain",
"Definition:Radius of Convergence/Real Domain",
"Definition:Infinite",
"Definition:Interval of Convergence"
] | [
"Nth Root Test",
"Definition:Convergent Series",
"Definition:Radius of Convergence/Real Domain"
] |
proofwiki-14861 | Comparison Test for Convergence of Power Series | Let $A = \ds \sum_{n \mathop \ge 0} a_n z^n$ and $B = \ds \sum_{n \mathop \ge 0} b_n z^n$ be power series in $\C$.
Let $R_A$ and $R_B$ be the radii of convergence of $A$ and $B$ respectively.
Let $\cmod {b_n} \le \cmod {a_n}$ for all $n \in \N$.
Then $R_A \le R_B$. | {{AimForCont}} $R_A > R_B$.
Let $z_0 \in \C$ such that $R_B < \cmod {z_0} < R_A$.
Then $A$ is convergent at $z_0$ but $B$ is divergent at $z_0$.
But by the Comparison Test, if $A$ is convergent at $z_0$ then $B$ is also convergent at $z_0$.
From this contradiction it follows that there can be no such $z_0$.
That is:
:$... | Let $A = \ds \sum_{n \mathop \ge 0} a_n z^n$ and $B = \ds \sum_{n \mathop \ge 0} b_n z^n$ be [[Definition:Complex Power Series|power series in $\C$]].
Let $R_A$ and $R_B$ be the [[Definition:Radius of Convergence|radii of convergence]] of $A$ and $B$ respectively.
Let $\cmod {b_n} \le \cmod {a_n}$ for all $n \in \N$... | {{AimForCont}} $R_A > R_B$.
Let $z_0 \in \C$ such that $R_B < \cmod {z_0} < R_A$.
Then $A$ is [[Definition:Convergent Complex Series|convergent]] at $z_0$ but $B$ is [[Definition:Divergent Series|divergent]] at $z_0$.
But by the [[Comparison Test]], if $A$ is [[Definition:Convergent Complex Series|convergent]] at $z... | Comparison Test for Convergence of Power Series | https://proofwiki.org/wiki/Comparison_Test_for_Convergence_of_Power_Series | https://proofwiki.org/wiki/Comparison_Test_for_Convergence_of_Power_Series | [
"Radius of Convergence",
"Complex Power Series"
] | [
"Definition:Power Series/Complex Domain",
"Definition:Radius of Convergence"
] | [
"Definition:Convergent Series/Number Field",
"Definition:Divergent Series",
"Comparison Test",
"Definition:Convergent Series/Number Field",
"Definition:Convergent Series/Number Field",
"Definition:Contradiction"
] |
proofwiki-14862 | Bounds for Modulus of e^z on Circle x^2 + y^2 - 2x - 2y - 2 = 0 | Consider the circle $C$ embedded in the complex plane defined by the equation:
:$x^2 + y^2 - 2 x - 2 y - 2 = 0$
Let $z = x + i y \in \C$ be a point lying on $C$.
Then:
:$e^{-1} \le \cmod {e^z} \le e^3$ | {{begin-eqn}}
{{eqn | l = x^2 + y^2 - 2 x - 2 y - 2
| r = 0
| c =
}}
{{eqn | ll= \leadsto
| l = \paren {\paren {x - 1}^2 - 1} + \paren {\paren {y - 1}^2 - 1} - 2
| r = 0
| c =
}}
{{eqn | ll= \leadsto
| l = \paren {x - 1}^2 + \paren {y - 1}^2
| r = 4
| c =
}}
{{end-eqn}... | Consider the [[Definition:Circle|circle]] $C$ embedded in the [[Definition:Complex Plane|complex plane]] defined by the equation:
:$x^2 + y^2 - 2 x - 2 y - 2 = 0$
Let $z = x + i y \in \C$ be a [[Definition:Point|point]] lying on $C$.
Then:
:$e^{-1} \le \cmod {e^z} \le e^3$ | {{begin-eqn}}
{{eqn | l = x^2 + y^2 - 2 x - 2 y - 2
| r = 0
| c =
}}
{{eqn | ll= \leadsto
| l = \paren {\paren {x - 1}^2 - 1} + \paren {\paren {y - 1}^2 - 1} - 2
| r = 0
| c =
}}
{{eqn | ll= \leadsto
| l = \paren {x - 1}^2 + \paren {y - 1}^2
| r = 4
| c =
}}
{{end-eqn}... | Bounds for Modulus of e^z on Circle x^2 + y^2 - 2x - 2y - 2 = 0 | https://proofwiki.org/wiki/Bounds_for_Modulus_of_e^z_on_Circle_x^2_+_y^2_-_2x_-_2y_-_2_=_0 | https://proofwiki.org/wiki/Bounds_for_Modulus_of_e^z_on_Circle_x^2_+_y^2_-_2x_-_2y_-_2_=_0 | [
"Circles",
"Geometry of Complex Plane",
"Equation of Circle"
] | [
"Definition:Circle",
"Definition:Complex Number/Complex Plane",
"Definition:Point"
] | [
"Definition:Circle",
"Definition:Circle/Center",
"Definition:Circle/Radius",
"Modulus of Exponential is Exponential of Real Part",
"Definition:Circle",
"Exponential is Strictly Increasing"
] |
proofwiki-14863 | Hyperbolic Cosine of Complex Number | Let $a$ and $b$ be real numbers.
Let $i$ be the imaginary unit.
Then:
:$\map \cosh {a + b i} = \cosh a \cos b + i \sinh a \sin b$
where:
:$\cos$ denotes the real cosine function
:$\sin$ denotes the real sine function
:$\sinh$ denotes the hyperbolic sine function
:$\cosh$ denotes the hyperbolic cosine function | {{begin-eqn}}
{{eqn | l = \map \cosh {a + b i}
| r = \cosh a \map \cosh {b i} + \sinh a \map \sinh {b i}
| c = Hyperbolic Cosine of Sum
}}
{{eqn | r = \cosh a \cos b + \sinh a \map \sinh {b i}
| c = Cosine in terms of Hyperbolic Cosine
}}
{{eqn | r = \cosh a \cos b + i \sinh a \sin b
| c = Sine ... | Let $a$ and $b$ be [[Definition:Real Number|real numbers]].
Let $i$ be the [[Definition:Imaginary Unit|imaginary unit]].
Then:
:$\map \cosh {a + b i} = \cosh a \cos b + i \sinh a \sin b$
where:
:$\cos$ denotes the [[Definition:Real Cosine Function|real cosine function]]
:$\sin$ denotes the [[Definition:Real Sine Fun... | {{begin-eqn}}
{{eqn | l = \map \cosh {a + b i}
| r = \cosh a \map \cosh {b i} + \sinh a \map \sinh {b i}
| c = [[Hyperbolic Cosine of Sum]]
}}
{{eqn | r = \cosh a \cos b + \sinh a \map \sinh {b i}
| c = [[Cosine in terms of Hyperbolic Cosine]]
}}
{{eqn | r = \cosh a \cos b + i \sinh a \sin b
| c... | Hyperbolic Cosine of Complex Number/Proof 1 | https://proofwiki.org/wiki/Hyperbolic_Cosine_of_Complex_Number | https://proofwiki.org/wiki/Hyperbolic_Cosine_of_Complex_Number/Proof_1 | [
"Hyperbolic Cosine Function",
"Complex Numbers",
"Hyperbolic Cosine of Complex Number"
] | [
"Definition:Real Number",
"Definition:Complex Number/Imaginary Unit",
"Definition:Cosine/Real Function",
"Definition:Sine/Real Function",
"Definition:Hyperbolic Sine",
"Definition:Hyperbolic Cosine"
] | [
"Hyperbolic Cosine of Sum",
"Cosine in terms of Hyperbolic Cosine",
"Sine in terms of Hyperbolic Sine"
] |
proofwiki-14864 | Hyperbolic Cosine of Complex Number | Let $a$ and $b$ be real numbers.
Let $i$ be the imaginary unit.
Then:
:$\map \cosh {a + b i} = \cosh a \cos b + i \sinh a \sin b$
where:
:$\cos$ denotes the real cosine function
:$\sin$ denotes the real sine function
:$\sinh$ denotes the hyperbolic sine function
:$\cosh$ denotes the hyperbolic cosine function | {{begin-eqn}}
{{eqn | l = \cosh a \cos b - i \sinh a \sin b
| r = \frac {e^a + e^{-a} } 2 \frac {e^{i b} + e^{-i b} } 2 + i \frac {e^a - e^{-a} } {2 i} \frac {e^{i b} - e^{-i b} } 2
| c = {{Defof|Hyperbolic Cosine}}, Euler's Cosine Identity, {{Defof|Hyperbolic Sine}}, Euler's Sine Identity
}}
{{eqn | r = \f... | Let $a$ and $b$ be [[Definition:Real Number|real numbers]].
Let $i$ be the [[Definition:Imaginary Unit|imaginary unit]].
Then:
:$\map \cosh {a + b i} = \cosh a \cos b + i \sinh a \sin b$
where:
:$\cos$ denotes the [[Definition:Real Cosine Function|real cosine function]]
:$\sin$ denotes the [[Definition:Real Sine Fun... | {{begin-eqn}}
{{eqn | l = \cosh a \cos b - i \sinh a \sin b
| r = \frac {e^a + e^{-a} } 2 \frac {e^{i b} + e^{-i b} } 2 + i \frac {e^a - e^{-a} } {2 i} \frac {e^{i b} - e^{-i b} } 2
| c = {{Defof|Hyperbolic Cosine}}, [[Euler's Cosine Identity]], {{Defof|Hyperbolic Sine}}, [[Euler's Sine Identity]]
}}
{{eqn ... | Hyperbolic Cosine of Complex Number/Proof 2 | https://proofwiki.org/wiki/Hyperbolic_Cosine_of_Complex_Number | https://proofwiki.org/wiki/Hyperbolic_Cosine_of_Complex_Number/Proof_2 | [
"Hyperbolic Cosine Function",
"Complex Numbers",
"Hyperbolic Cosine of Complex Number"
] | [
"Definition:Real Number",
"Definition:Complex Number/Imaginary Unit",
"Definition:Cosine/Real Function",
"Definition:Sine/Real Function",
"Definition:Hyperbolic Sine",
"Definition:Hyperbolic Cosine"
] | [
"Euler's Cosine Identity",
"Euler's Sine Identity"
] |
proofwiki-14865 | Hyperbolic Tangent of Complex Number/Formulation 1 | :$\tanh \paren {a + b i} = \dfrac {\sinh a \cos b + i \cosh a \sin b} {\cosh a \cos b + i \sinh a \sin b}$ | {{begin-eqn}}
{{eqn | l = \tanh \paren {a + b i}
| r = \frac {\sinh \paren {a + b i} } {\cosh \paren {a + b i} }
| c = {{Defof|Hyperbolic Tangent}}
}}
{{eqn | r = \dfrac {\sinh a \cos b + i \cosh a \sin b} {\cosh a \cos b + i \sinh a \sin b}
| c = Hyperbolic Sine of Complex Number and Hyperbolic Cosin... | :$\tanh \paren {a + b i} = \dfrac {\sinh a \cos b + i \cosh a \sin b} {\cosh a \cos b + i \sinh a \sin b}$ | {{begin-eqn}}
{{eqn | l = \tanh \paren {a + b i}
| r = \frac {\sinh \paren {a + b i} } {\cosh \paren {a + b i} }
| c = {{Defof|Hyperbolic Tangent}}
}}
{{eqn | r = \dfrac {\sinh a \cos b + i \cosh a \sin b} {\cosh a \cos b + i \sinh a \sin b}
| c = [[Hyperbolic Sine of Complex Number]] and [[Hyperbolic... | Hyperbolic Tangent of Complex Number/Formulation 1 | https://proofwiki.org/wiki/Hyperbolic_Tangent_of_Complex_Number/Formulation_1 | https://proofwiki.org/wiki/Hyperbolic_Tangent_of_Complex_Number/Formulation_1 | [
"Hyperbolic Tangent of Complex Number"
] | [] | [
"Hyperbolic Sine of Complex Number",
"Hyperbolic Cosine of Complex Number"
] |
proofwiki-14866 | Hyperbolic Tangent of Complex Number/Formulation 2 | :$\tanh \paren {a + b i} = \dfrac {\tanh a + i \tan b} {1 + i \tanh a \tan b}$ | {{begin-eqn}}
{{eqn | l = \tanh \paren {a + b i}
| r = \dfrac {\sinh a \cos b + i \cosh a \sin b} {\cosh a \cos b + i \sinh a \sin b}
| c = Hyperbolic Tangent of Complex Number: Formulation 1
}}
{{eqn | r = \dfrac {\tanh a \cos b + i \sin b} {\cos b + i \tanh a \sin b}
| c = multiplying denominator an... | :$\tanh \paren {a + b i} = \dfrac {\tanh a + i \tan b} {1 + i \tanh a \tan b}$ | {{begin-eqn}}
{{eqn | l = \tanh \paren {a + b i}
| r = \dfrac {\sinh a \cos b + i \cosh a \sin b} {\cosh a \cos b + i \sinh a \sin b}
| c = [[Hyperbolic Tangent of Complex Number/Formulation 1|Hyperbolic Tangent of Complex Number: Formulation 1]]
}}
{{eqn | r = \dfrac {\tanh a \cos b + i \sin b} {\cos b + i... | Hyperbolic Tangent of Complex Number/Formulation 2 | https://proofwiki.org/wiki/Hyperbolic_Tangent_of_Complex_Number/Formulation_2 | https://proofwiki.org/wiki/Hyperbolic_Tangent_of_Complex_Number/Formulation_2 | [
"Hyperbolic Tangent of Complex Number"
] | [] | [
"Hyperbolic Tangent of Complex Number/Formulation 1",
"Definition:Fraction/Denominator",
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Definition:Fraction/Numerator"
] |
proofwiki-14867 | Hyperbolic Tangent of Complex Number/Formulation 3 | :$\tanh \paren {a + b i} = \dfrac {\tanh a + \tanh a \tan^2 b} {1 + \tanh^2 a \tan^2 b} + \dfrac {\tan b - \tanh^2 a \tan b} {1 + \tanh^2 a \tan^2 b} i$ | {{begin-eqn}}
{{eqn | l = \tan \paren {a + b i}
| r = \dfrac {\tanh a + i \tan b} {1 + i \tanh a \tan b}
| c = Hyperbolic Tangent of Complex Number: Formulation 2
}}
{{eqn | r = \frac {\paren {\tanh a + i \tan b} \paren {1 - i \tanh a \tan b} } {1 + \tanh^2 a \tan^2 b}
| c = multiplying denominator an... | :$\tanh \paren {a + b i} = \dfrac {\tanh a + \tanh a \tan^2 b} {1 + \tanh^2 a \tan^2 b} + \dfrac {\tan b - \tanh^2 a \tan b} {1 + \tanh^2 a \tan^2 b} i$ | {{begin-eqn}}
{{eqn | l = \tan \paren {a + b i}
| r = \dfrac {\tanh a + i \tan b} {1 + i \tanh a \tan b}
| c = [[Hyperbolic Tangent of Complex Number/Formulation 2|Hyperbolic Tangent of Complex Number: Formulation 2]]
}}
{{eqn | r = \frac {\paren {\tanh a + i \tan b} \paren {1 - i \tanh a \tan b} } {1 + \ta... | Hyperbolic Tangent of Complex Number/Formulation 3 | https://proofwiki.org/wiki/Hyperbolic_Tangent_of_Complex_Number/Formulation_3 | https://proofwiki.org/wiki/Hyperbolic_Tangent_of_Complex_Number/Formulation_3 | [
"Hyperbolic Tangent of Complex Number"
] | [] | [
"Hyperbolic Tangent of Complex Number/Formulation 2",
"Definition:Fraction/Denominator",
"Definition:Fraction/Numerator"
] |
proofwiki-14868 | Hyperbolic Cosecant of Complex Number | Let $a$ and $b$ be real numbers.
Let $i$ be the imaginary unit.
Then:
:$\map \csch {a + b i} = \dfrac {\sinh a \cos b - i \cosh a \sin b} {\sinh^2 a \cos^2 b + \cosh^2 a \sin^2 b}$
where:
:$\csch$ denotes the hyperbolic cosecant function.
:$\sin$ denotes the real sine function
:$\cos$ denotes the real cosine function
:... | {{begin-eqn}}
{{eqn | l = \map \csch {a + b i}
| r = \frac 1 {\map \sinh {a + b i} }
| c = {{Defof|Hyperbolic Cosecant}}
}}
{{eqn | r = \dfrac 1 {\sinh a \cos b + i \cosh a \sin b}
| c = Hyperbolic Sine of Complex Number
}}
{{eqn | r = \dfrac {\sinh a \cos b - i \cosh a \sin b} {\paren {\sinh a \cos b... | Let $a$ and $b$ be [[Definition:Real Number|real numbers]].
Let $i$ be the [[Definition:Imaginary Unit|imaginary unit]].
Then:
:$\map \csch {a + b i} = \dfrac {\sinh a \cos b - i \cosh a \sin b} {\sinh^2 a \cos^2 b + \cosh^2 a \sin^2 b}$
where:
:$\csch$ denotes the [[Definition:Hyperbolic Cosecant|hyperbolic coseca... | {{begin-eqn}}
{{eqn | l = \map \csch {a + b i}
| r = \frac 1 {\map \sinh {a + b i} }
| c = {{Defof|Hyperbolic Cosecant}}
}}
{{eqn | r = \dfrac 1 {\sinh a \cos b + i \cosh a \sin b}
| c = [[Hyperbolic Sine of Complex Number]]
}}
{{eqn | r = \dfrac {\sinh a \cos b - i \cosh a \sin b} {\paren {\sinh a \c... | Hyperbolic Cosecant of Complex Number | https://proofwiki.org/wiki/Hyperbolic_Cosecant_of_Complex_Number | https://proofwiki.org/wiki/Hyperbolic_Cosecant_of_Complex_Number | [
"Hyperbolic Cosecant Function",
"Complex Numbers"
] | [
"Definition:Real Number",
"Definition:Complex Number/Imaginary Unit",
"Definition:Hyperbolic Cosecant",
"Definition:Sine/Real Function",
"Definition:Cosine/Real Function",
"Definition:Hyperbolic Sine",
"Definition:Hyperbolic Cosine"
] | [
"Hyperbolic Sine of Complex Number",
"Definition:Fraction/Denominator",
"Definition:Fraction/Numerator",
"Difference of Two Squares"
] |
proofwiki-14869 | Hyperbolic Secant of Complex Number | Let $a$ and $b$ be real numbers.
Let $i$ be the imaginary unit.
Then:
:$\sech \paren {a + b i} = \dfrac {\cosh a \cos b - i \sinh a \sin b} {\cosh^2 a \cos^2 b + \sinh^2 a \sin^2 b}$
where:
:$\sech$ denotes the hyperbolic secant function.
:$\sin$ denotes the real sine function
:$\cos$ denotes the real cosine function
:... | {{begin-eqn}}
{{eqn | l = \sech \paren {a + b i}
| r = \frac 1 {\cosh \paren {a + b i} }
| c = {{Defof|Hyperbolic Secant}}
}}
{{eqn | r = \dfrac 1 {\cosh a \cos b + i \sinh a \sin b}
| c = Hyperbolic Cosine of Complex Number
}}
{{eqn | r = \dfrac {\cosh a \cos b - i \sinh a \sin b} {\paren {\cosh a \c... | Let $a$ and $b$ be [[Definition:Real Number|real numbers]].
Let $i$ be the [[Definition:Imaginary Unit|imaginary unit]].
Then:
:$\sech \paren {a + b i} = \dfrac {\cosh a \cos b - i \sinh a \sin b} {\cosh^2 a \cos^2 b + \sinh^2 a \sin^2 b}$
where:
:$\sech$ denotes the [[Definition:Hyperbolic Secant|hyperbolic secant... | {{begin-eqn}}
{{eqn | l = \sech \paren {a + b i}
| r = \frac 1 {\cosh \paren {a + b i} }
| c = {{Defof|Hyperbolic Secant}}
}}
{{eqn | r = \dfrac 1 {\cosh a \cos b + i \sinh a \sin b}
| c = [[Hyperbolic Cosine of Complex Number]]
}}
{{eqn | r = \dfrac {\cosh a \cos b - i \sinh a \sin b} {\paren {\cosh ... | Hyperbolic Secant of Complex Number | https://proofwiki.org/wiki/Hyperbolic_Secant_of_Complex_Number | https://proofwiki.org/wiki/Hyperbolic_Secant_of_Complex_Number | [
"Hyperbolic Secant Function",
"Complex Numbers"
] | [
"Definition:Real Number",
"Definition:Complex Number/Imaginary Unit",
"Definition:Hyperbolic Secant",
"Definition:Sine/Real Function",
"Definition:Cosine/Real Function",
"Definition:Hyperbolic Sine",
"Definition:Hyperbolic Cosine"
] | [
"Hyperbolic Cosine of Complex Number",
"Definition:Fraction/Denominator",
"Definition:Fraction/Numerator",
"Difference of Two Squares"
] |
proofwiki-14870 | Hyperbolic Cotangent of Complex Number/Formulation 2 | :$\map \coth {a + b i} = \dfrac {1 - i \coth a \cot b} {\coth a - i \cot b}$ | {{begin-eqn}}
{{eqn | l = \map \coth {a + b i}
| r = \dfrac {\cosh a \cos b + i \sinh a \sin b} {\sinh a \cos b + i \cosh a \sin b}
| c = Hyperbolic Cotangent of Complex Number: Formulation 1
}}
{{eqn | r = \dfrac {\coth a \cos b + i \sin b} {\cos b + i \coth a \sin b}
| c = multiplying denominator an... | :$\map \coth {a + b i} = \dfrac {1 - i \coth a \cot b} {\coth a - i \cot b}$ | {{begin-eqn}}
{{eqn | l = \map \coth {a + b i}
| r = \dfrac {\cosh a \cos b + i \sinh a \sin b} {\sinh a \cos b + i \cosh a \sin b}
| c = [[Hyperbolic Cotangent of Complex Number/Formulation 1|Hyperbolic Cotangent of Complex Number: Formulation 1]]
}}
{{eqn | r = \dfrac {\coth a \cos b + i \sin b} {\cos b +... | Hyperbolic Cotangent of Complex Number/Formulation 2 | https://proofwiki.org/wiki/Hyperbolic_Cotangent_of_Complex_Number/Formulation_2 | https://proofwiki.org/wiki/Hyperbolic_Cotangent_of_Complex_Number/Formulation_2 | [
"Hyperbolic Cotangent of Complex Number"
] | [] | [
"Hyperbolic Cotangent of Complex Number/Formulation 1",
"Definition:Fraction/Denominator",
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Definition:Fraction/Numerator"
] |
proofwiki-14871 | Hyperbolic Cotangent of Complex Number/Formulation 3 | :$\map \coth {a + b i} = \dfrac {\coth a + \coth a \cot^2 b} {\coth^2 a + \cot^2 b} + \dfrac {\cot b - \coth^2 a \cot b} {\coth^2 a + \cot^2 b} i$ | {{begin-eqn}}
{{eqn | l = \map \coth {a + b i}
| r = \dfrac {1 - i \coth a \cot b} {\coth a - i \cot b}
| c = Hyperbolic Cotangent of Complex Number: Formulation 2
}}
{{eqn | r = \dfrac {\paren {1 - i \coth a \cot b} \paren {\coth a + i \cot b} } {\paren {\coth a - i \cot b} \paren {\coth a + i \cot b} }
... | :$\map \coth {a + b i} = \dfrac {\coth a + \coth a \cot^2 b} {\coth^2 a + \cot^2 b} + \dfrac {\cot b - \coth^2 a \cot b} {\coth^2 a + \cot^2 b} i$ | {{begin-eqn}}
{{eqn | l = \map \coth {a + b i}
| r = \dfrac {1 - i \coth a \cot b} {\coth a - i \cot b}
| c = [[Hyperbolic Cotangent of Complex Number/Formulation 2|Hyperbolic Cotangent of Complex Number: Formulation 2]]
}}
{{eqn | r = \dfrac {\paren {1 - i \coth a \cot b} \paren {\coth a + i \cot b} } {\pa... | Hyperbolic Cotangent of Complex Number/Formulation 3 | https://proofwiki.org/wiki/Hyperbolic_Cotangent_of_Complex_Number/Formulation_3 | https://proofwiki.org/wiki/Hyperbolic_Cotangent_of_Complex_Number/Formulation_3 | [
"Hyperbolic Cotangent of Complex Number"
] | [] | [
"Hyperbolic Cotangent of Complex Number/Formulation 2",
"Definition:Fraction/Denominator",
"Definition:Fraction/Numerator",
"Difference of Two Squares"
] |
proofwiki-14872 | Hyperbolic Tangent of Complex Number/Formulation 4 | :$\map \tanh {a + b i} = \dfrac {\sinh 2 a + i \sin 2 b} {\cosh 2 a + \cos 2 b}$ | {{begin-eqn}}
{{eqn | l = \map \tanh {a + b i}
| r = \dfrac {\sinh a \cos b + i \cosh a \sin b} {\cosh a \cos b + i \sinh a \sin b}
| c = Hyperbolic Tangent of Complex Number: Formulation 1
}}
{{eqn | r = \dfrac {\paren {\sinh a \cos b + i \cosh a \sin b} \paren {\cosh a \cos b - i \sinh a \sin b} } {\paren... | :$\map \tanh {a + b i} = \dfrac {\sinh 2 a + i \sin 2 b} {\cosh 2 a + \cos 2 b}$ | {{begin-eqn}}
{{eqn | l = \map \tanh {a + b i}
| r = \dfrac {\sinh a \cos b + i \cosh a \sin b} {\cosh a \cos b + i \sinh a \sin b}
| c = [[Hyperbolic Tangent of Complex Number/Formulation 1|Hyperbolic Tangent of Complex Number: Formulation 1]]
}}
{{eqn | r = \dfrac {\paren {\sinh a \cos b + i \cosh a \sin ... | Hyperbolic Tangent of Complex Number/Formulation 4 | https://proofwiki.org/wiki/Hyperbolic_Tangent_of_Complex_Number/Formulation_4 | https://proofwiki.org/wiki/Hyperbolic_Tangent_of_Complex_Number/Formulation_4 | [
"Hyperbolic Tangent of Complex Number"
] | [] | [
"Hyperbolic Tangent of Complex Number/Formulation 1",
"Sum of Squares of Sine and Cosine",
"Difference of Squares of Hyperbolic Cosine and Sine",
"Double Angle Formulas/Hyperbolic Sine",
"Double Angle Formulas/Sine",
"Double Angle Formulas/Cosine",
"Double Angle Formulas/Hyperbolic Cosine",
"Differenc... |
proofwiki-14873 | Tangent of Complex Number/Formulation 4 | :$\tan \paren {a + b i} = \dfrac {\sin 2 a + i \sinh 2 b} {\cos 2 a + \cosh 2 b}$ | {{begin-eqn}}
{{eqn | l = \tan \paren {a + b i}
| r = \dfrac {\sin a \cosh b + i \cos a \sinh b} {\cos a \cosh b - i \sin a \sinh b}
| c = Tangent of Complex Number: Formulation 1
}}
{{eqn | r = \dfrac {\paren {\sin a \cosh b + i \cos a \sinh b} \paren {\cos a \cosh b + i \sin a \sinh b} } {\paren {\cos a \... | :$\tan \paren {a + b i} = \dfrac {\sin 2 a + i \sinh 2 b} {\cos 2 a + \cosh 2 b}$ | {{begin-eqn}}
{{eqn | l = \tan \paren {a + b i}
| r = \dfrac {\sin a \cosh b + i \cos a \sinh b} {\cos a \cosh b - i \sin a \sinh b}
| c = [[Tangent of Complex Number/Formulation 1|Tangent of Complex Number: Formulation 1]]
}}
{{eqn | r = \dfrac {\paren {\sin a \cosh b + i \cos a \sinh b} \paren {\cos a \co... | Tangent of Complex Number/Formulation 4 | https://proofwiki.org/wiki/Tangent_of_Complex_Number/Formulation_4 | https://proofwiki.org/wiki/Tangent_of_Complex_Number/Formulation_4 | [
"Tangent of Complex Number"
] | [] | [
"Tangent of Complex Number/Formulation 1",
"Difference of Squares of Hyperbolic Cosine and Sine",
"Sum of Squares of Sine and Cosine",
"Double Angle Formulas/Sine",
"Double Angle Formulas/Hyperbolic Sine",
"Double Angle Formulas/Cosine",
"Double Angle Formulas/Hyperbolic Cosine",
"Difference of Square... |
proofwiki-14874 | Characterisation of Cauchy Sequence in Non-Archimedean Norm | Let $\struct {R, \norm {\, \cdot \,} }$ be a normed division ring with non-Archimedean norm $\norm {\, \cdot \,}$.
Let $\sequence {x_n}$ be a sequence in $R$.
Then:
:$\sequence {x_n}$ is a Cauchy sequence
{{iff}}:
:$\ds \lim_{n \mathop \to \infty} \norm {x_{n + 1} - x_n} = 0$ | === Necessary Condition ===
{{:Characterisation of Cauchy Sequence in Non-Archimedean Norm/Necessary Condition}}{{qed|lemma}} | Let $\struct {R, \norm {\, \cdot \,} }$ be a [[Definition:Normed Division Ring |normed division ring]] with [[Definition:Non-Archimedean Division Ring Norm|non-Archimedean norm]] $\norm {\, \cdot \,}$.
Let $\sequence {x_n}$ be a [[Definition:Sequence|sequence]] in $R$.
Then:
:$\sequence {x_n}$ is a [[Definition:Cauch... | === [[Characterisation of Cauchy Sequence in Non-Archimedean Norm/Necessary Condition|Necessary Condition]] ===
{{:Characterisation of Cauchy Sequence in Non-Archimedean Norm/Necessary Condition}}{{qed|lemma}} | Characterisation of Cauchy Sequence in Non-Archimedean Norm | https://proofwiki.org/wiki/Characterisation_of_Cauchy_Sequence_in_Non-Archimedean_Norm | https://proofwiki.org/wiki/Characterisation_of_Cauchy_Sequence_in_Non-Archimedean_Norm | [
"Normed Division Rings",
"Cauchy Sequences",
"Non-Archimedean Norms",
"Cauchy Sequences in Normed Division Rings",
"Characterisation of Cauchy Sequence in Non-Archimedean Norm"
] | [
"Definition:Normed Division Ring ",
"Definition:Non-Archimedean/Norm (Division Ring)",
"Definition:Sequence",
"Definition:Cauchy Sequence/Normed Division Ring"
] | [
"Characterisation of Cauchy Sequence in Non-Archimedean Norm/Necessary Condition"
] |
proofwiki-14875 | Characterisation of Cauchy Sequence in Non-Archimedean Norm/Necessary Condition | Let $\struct {R, \norm {\,\cdot\,} }$ be a normed division ring with non-Archimedean norm $\norm {\,\cdot\,}$.
Let $\sequence {x_n}$ be a Cauchy sequence in $R$.
Then:
:$\lim_{n \mathop \to \infty} \norm {x_{n + 1} - x_n} = 0$ | Let $\epsilon > 0$ be given.
By the definition of a Cauchy sequence:
:$\exists N: \forall n, m > N: \norm {x_n - x_m} < \epsilon$
So
:$\exists N: \forall n > N: \norm {x_{n + 1} - x_n} < \epsilon$
Hence the result follows:
:$\lim_{n \mathop \to \infty} \norm {x_{n + 1} - x_n} = 0$. | Let $\struct {R, \norm {\,\cdot\,} }$ be a [[Definition:Normed Division Ring|normed division ring]] with [[Definition:Non-Archimedean Division Ring Norm|non-Archimedean norm]] $\norm {\,\cdot\,}$.
Let $\sequence {x_n}$ be a [[Definition:Cauchy Sequence in Normed Division Ring|Cauchy sequence]] in $R$.
Then:
:$\lim_{... | Let $\epsilon > 0$ be given.
By the definition of a [[Definition:Cauchy Sequence in Normed Division Ring|Cauchy sequence]]:
:$\exists N: \forall n, m > N: \norm {x_n - x_m} < \epsilon$
So
:$\exists N: \forall n > N: \norm {x_{n + 1} - x_n} < \epsilon$
Hence the result follows:
:$\lim_{n \mathop \to \infty} \norm ... | Characterisation of Cauchy Sequence in Non-Archimedean Norm/Necessary Condition | https://proofwiki.org/wiki/Characterisation_of_Cauchy_Sequence_in_Non-Archimedean_Norm/Necessary_Condition | https://proofwiki.org/wiki/Characterisation_of_Cauchy_Sequence_in_Non-Archimedean_Norm/Necessary_Condition | [
"Characterisation of Cauchy Sequence in Non-Archimedean Norm"
] | [
"Definition:Normed Division Ring",
"Definition:Non-Archimedean/Norm (Division Ring)",
"Definition:Cauchy Sequence/Normed Division Ring"
] | [
"Definition:Cauchy Sequence/Normed Division Ring"
] |
proofwiki-14876 | Characterisation of Cauchy Sequence in Non-Archimedean Norm/Sufficient Condition | Let $\struct {R, \norm {\,\cdot\,} }$ be a normed division ring with non-Archimedean norm $\norm {\,\cdot\,}$.
Let $\sequence {x_n}$ be a sequence in $R$.
Let $\ds \lim_{n \mathop \to \infty} \norm {x_{n + 1} - x_n} = 0$.
Then:
:$\sequence {x_n}$ is a Cauchy sequence. | Let $\epsilon > 0$ be given.
By assumption $\exists N \in \N$ such that:
:$(1) \quad \forall n > N: \norm {x_{n + 1} - x_n} < 0$
Suppose $n, m > N$, and $m = n + r > n$.
Then:
{{begin-eqn}}
{{eqn | l = \norm {x_m - x_n}
| r = \norm {x_{n + r} - x_{n + r - 1} + x_{n + r - 1} - x_{n + r - 2} + \dotsb + x_{n + 1} -... | Let $\struct {R, \norm {\,\cdot\,} }$ be a [[Definition:Normed Division Ring|normed division ring]] with [[Definition:Non-Archimedean Division Ring Norm|non-Archimedean norm]] $\norm {\,\cdot\,}$.
Let $\sequence {x_n}$ be a [[Definition:Sequence|sequence]] in $R$.
Let $\ds \lim_{n \mathop \to \infty} \norm {x_{n + 1}... | Let $\epsilon > 0$ be given.
By assumption $\exists N \in \N$ such that:
:$(1) \quad \forall n > N: \norm {x_{n + 1} - x_n} < 0$
Suppose $n, m > N$, and $m = n + r > n$.
Then:
{{begin-eqn}}
{{eqn | l = \norm {x_m - x_n}
| r = \norm {x_{n + r} - x_{n + r - 1} + x_{n + r - 1} - x_{n + r - 2} + \dotsb + x_{n +... | Characterisation of Cauchy Sequence in Non-Archimedean Norm/Sufficient Condition | https://proofwiki.org/wiki/Characterisation_of_Cauchy_Sequence_in_Non-Archimedean_Norm/Sufficient_Condition | https://proofwiki.org/wiki/Characterisation_of_Cauchy_Sequence_in_Non-Archimedean_Norm/Sufficient_Condition | [
"Characterisation of Cauchy Sequence in Non-Archimedean Norm"
] | [
"Definition:Normed Division Ring",
"Definition:Non-Archimedean/Norm (Division Ring)",
"Definition:Sequence",
"Definition:Cauchy Sequence/Normed Division Ring"
] | [
"Definition:Non-Archimedean/Norm (Division Ring)",
"Definition:Cauchy Sequence/Normed Division Ring"
] |
proofwiki-14877 | Power Series of Sine of Odd Theta | Let $r \in \R$ such that $\size r < 1$.
Let $\theta \in \R$ such that $\theta \ne m \pi$ for any $m \in \Z$.
Then:
{{begin-eqn}}
{{eqn | l = \sum_{k \mathop \ge 0} \map \sin {2 k + 1} \theta r^k
| r = \sin \theta + r \sin 3 \theta + r^2 \sin 5 \theta + \cdots
| c =
}}
{{eqn | r = \dfrac {\paren {1 + r} \si... | From Euler's Formula:
:$\map \exp {i \theta} = \cos \theta + i \sin \theta$
Hence:
{{begin-eqn}}
{{eqn | l = \sum_{k \mathop = 0}^\infty \map \sin {2 k + 1} \theta r^k
| r = \map \Im {\sum_{k \mathop = 0}^\infty r^k \map \exp {\paren {2 k + 1} i \theta} }
| c =
}}
{{eqn | r = \map \Im {\map \exp {i \theta}... | Let $r \in \R$ such that $\size r < 1$.
Let $\theta \in \R$ such that $\theta \ne m \pi$ for any $m \in \Z$.
Then:
{{begin-eqn}}
{{eqn | l = \sum_{k \mathop \ge 0} \map \sin {2 k + 1} \theta r^k
| r = \sin \theta + r \sin 3 \theta + r^2 \sin 5 \theta + \cdots
| c =
}}
{{eqn | r = \dfrac {\paren {1 + r} \... | From [[Euler's Formula]]:
:$\map \exp {i \theta} = \cos \theta + i \sin \theta$
Hence:
{{begin-eqn}}
{{eqn | l = \sum_{k \mathop = 0}^\infty \map \sin {2 k + 1} \theta r^k
| r = \map \Im {\sum_{k \mathop = 0}^\infty r^k \map \exp {\paren {2 k + 1} i \theta} }
| c =
}}
{{eqn | r = \map \Im {\map \exp {i \... | Power Series of Sine of Odd Theta | https://proofwiki.org/wiki/Power_Series_of_Sine_of_Odd_Theta | https://proofwiki.org/wiki/Power_Series_of_Sine_of_Odd_Theta | [
"Sine Function",
"Trigonometric Series"
] | [] | [
"Euler's Formula",
"Sum of Infinite Geometric Sequence",
"Definition:Fraction/Denominator",
"Definition:Fraction/Numerator",
"Euler's Cosine Identity",
"Euler's Formula",
"Definition:Complex Number/Imaginary Part",
"Sine of Difference"
] |
proofwiki-14878 | Sum of Infinite Series of Product of nth Power of cos 2 theta by 2n+1th Multiple of Sine | Let $\theta \in \R$ such that $\theta \ne m \pi$ for any $m \in \Z$.
Then:
{{begin-eqn}}
{{eqn | l = \sum_{n \mathop = 0}^\infty \paren {\cos 2 \theta}^n \sin \paren {2 n + 1} \theta
| r = \sin \theta + \cos 2 \theta \sin 3 \theta + \paren {\cos 2 \theta}^2 \sin 5 \theta + \paren {\cos 2 \theta}^3 \sin 7 \theta +... | Let $\theta \ne \dfrac {m \pi} 2$ for any $m \in \Z$.
Then $\size {\cos 2 \theta} < 1$.
{{begin-eqn}}
{{eqn | l = \sum_{k \mathop \ge 0} \sin \paren {2 k + 1} \theta r^k
| r = \dfrac {\paren {1 + r} \sin \theta} {1 - 2 r \cos 2 \theta + r^2}
| c = Power Series of Sine of Odd Theta: $\size r < 1$
}}
{{eqn | ... | Let $\theta \in \R$ such that $\theta \ne m \pi$ for any $m \in \Z$.
Then:
{{begin-eqn}}
{{eqn | l = \sum_{n \mathop = 0}^\infty \paren {\cos 2 \theta}^n \sin \paren {2 n + 1} \theta
| r = \sin \theta + \cos 2 \theta \sin 3 \theta + \paren {\cos 2 \theta}^2 \sin 5 \theta + \paren {\cos 2 \theta}^3 \sin 7 \thet... | Let $\theta \ne \dfrac {m \pi} 2$ for any $m \in \Z$.
Then $\size {\cos 2 \theta} < 1$.
{{begin-eqn}}
{{eqn | l = \sum_{k \mathop \ge 0} \sin \paren {2 k + 1} \theta r^k
| r = \dfrac {\paren {1 + r} \sin \theta} {1 - 2 r \cos 2 \theta + r^2}
| c = [[Power Series of Sine of Odd Theta]]: $\size r < 1$
}}
{{... | Sum of Infinite Series of Product of nth Power of cos 2 theta by 2n+1th Multiple of Sine | https://proofwiki.org/wiki/Sum_of_Infinite_Series_of_Product_of_nth_Power_of_cos_2_theta_by_2n+1th_Multiple_of_Sine | https://proofwiki.org/wiki/Sum_of_Infinite_Series_of_Product_of_nth_Power_of_cos_2_theta_by_2n+1th_Multiple_of_Sine | [
"Cosine Function"
] | [] | [
"Power Series of Sine of Odd Theta",
"Double Angle Formulas/Cosine",
"Sum of Squares of Sine and Cosine"
] |
proofwiki-14879 | Modulus of Limit/Normed Division Ring | Let $\struct {R, \norm {\, \cdot \,} }$ be a normed division ring.
Let $\sequence {x_n}$ be a convergent sequence in $R$ to the limit $l$.
That is, let $\ds \lim_{n \mathop \to \infty} x_n = l$.
Then
:$\ds \lim_{n \mathop \to \infty} \norm {x_n} = \norm l$ | By the Reverse Triangle Inequality, we have:
:$\cmod {\norm {x_n} - \norm l} \le \norm {x_n - l}$
Hence by the Squeeze Theorem and Convergent Sequence Minus Limit, $\norm {x_n} \to \norm l$ as $n \to \infty$.
{{Qed}} | Let $\struct {R, \norm {\, \cdot \,} }$ be a [[Definition:Normed Division Ring|normed division ring]].
Let $\sequence {x_n}$ be a [[Definition:Convergent Sequence in Normed Division Ring|convergent sequence]] in $R$ to the [[Definition:Limit of Sequence (Number Field)|limit]] $l$.
That is, let $\ds \lim_{n \mathop \t... | By the [[Reverse Triangle Inequality]], we have:
:$\cmod {\norm {x_n} - \norm l} \le \norm {x_n - l}$
Hence by the [[Squeeze Theorem]] and [[Convergent Sequence Minus Limit]], $\norm {x_n} \to \norm l$ as $n \to \infty$.
{{Qed}} | Modulus of Limit/Normed Division Ring | https://proofwiki.org/wiki/Modulus_of_Limit/Normed_Division_Ring | https://proofwiki.org/wiki/Modulus_of_Limit/Normed_Division_Ring | [
"Limits of Sequences",
"Modulus of Limit",
"Normed Division Rings",
"Modulus of Limit",
"Normed Division Rings"
] | [
"Definition:Normed Division Ring",
"Definition:Convergent Sequence/Normed Division Ring",
"Definition:Limit of Sequence (Number Field)"
] | [
"Reverse Triangle Inequality",
"Squeeze Theorem",
"Convergent Sequence Minus Limit"
] |
proofwiki-14880 | Three Points in Ultrametric Space have Two Equal Distances | Let $\struct {X, d}$ be an ultrametric space.
Let $x, y, z \in X$ with $\map d {x, z} \ne \map d {y, z}$.
Then:
:$\map d {x, y} = \max \set {\map d {x, z}, \map d {y, z} }$ | {{WLOG}}, let $\map d {x, z} > \map d {y, z}$.
Then:
{{begin-eqn}}
{{eqn | l = \map d {x, y}
| o = \le
| r = \max \set {\map d {x, z}, \map d {y, z} }
| c = {{Defof|Non-Archimedean Metric}}
}}
{{eqn | r = \map d {x, z}
| c = since $\map d {x, z} > \map d {y, z}$
}}
{{end-eqn}}
On the other hand:... | Let $\struct {X, d}$ be an [[Definition:Ultrametric Space|ultrametric space]].
Let $x, y, z \in X$ with $\map d {x, z} \ne \map d {y, z}$.
Then:
:$\map d {x, y} = \max \set {\map d {x, z}, \map d {y, z} }$ | {{WLOG}}, let $\map d {x, z} > \map d {y, z}$.
Then:
{{begin-eqn}}
{{eqn | l = \map d {x, y}
| o = \le
| r = \max \set {\map d {x, z}, \map d {y, z} }
| c = {{Defof|Non-Archimedean Metric}}
}}
{{eqn | r = \map d {x, z}
| c = since $\map d {x, z} > \map d {y, z}$
}}
{{end-eqn}}
On the other h... | Three Points in Ultrametric Space have Two Equal Distances | https://proofwiki.org/wiki/Three_Points_in_Ultrametric_Space_have_Two_Equal_Distances | https://proofwiki.org/wiki/Three_Points_in_Ultrametric_Space_have_Two_Equal_Distances | [
"Three Points in Ultrametric Space have Two Equal Distances",
"Ultrametric Spaces"
] | [
"Definition:Ultrametric Space"
] | [] |
proofwiki-14881 | Three Points in Ultrametric Space have Two Equal Distances/Corollary | Let $\struct {X, d}$ be an ultrametric space.
Let $x, y, z \in X$.
Then:
:at least two of the distances $\map d {x, y}$, $\map d {x, z}$ and $\map d {y, z}$ are equal. | Either:
:$\map d {x, z} = \map d {y, z}$
or:
:$\map d {x, z} \ne \map d {y, z}$
By Three Points in Ultrametric Space have Two Equal Distances:
:$\map d {x, z} = \map d {y, z}$ or $\map d {x, y} = \max \set {\map d {x, z}, \map d {y, z} }$
In either case two of the distances are equal.
{{Qed}} | Let $\struct {X, d}$ be an [[Definition:Ultrametric Space|ultrametric space]].
Let $x, y, z \in X$.
Then:
:at least two of the distances $\map d {x, y}$, $\map d {x, z}$ and $\map d {y, z}$ are equal. | Either:
:$\map d {x, z} = \map d {y, z}$
or:
:$\map d {x, z} \ne \map d {y, z}$
By [[Three Points in Ultrametric Space have Two Equal Distances]]:
:$\map d {x, z} = \map d {y, z}$ or $\map d {x, y} = \max \set {\map d {x, z}, \map d {y, z} }$
In either case two of the distances are equal.
{{Qed}} | Three Points in Ultrametric Space have Two Equal Distances/Corollary | https://proofwiki.org/wiki/Three_Points_in_Ultrametric_Space_have_Two_Equal_Distances/Corollary | https://proofwiki.org/wiki/Three_Points_in_Ultrametric_Space_have_Two_Equal_Distances/Corollary | [
"Three Points in Ultrametric Space have Two Equal Distances"
] | [
"Definition:Ultrametric Space"
] | [
"Three Points in Ultrametric Space have Two Equal Distances"
] |
proofwiki-14882 | Non-Archimedean Norm iff Non-Archimedean Metric/Necessary Condition | Let $\struct {R, \norm {\,\cdot\,}}$ be a non-Archimedean normed division ring.
Let $d$ be the metric induced by $\norm {\,\cdot\,}$.
Then $d$ is a non-Archimedean metric. | Let $x, y, z \in R$.
{{begin-eqn}}
{{eqn | l = \map d {x, y}
| r = \norm {x - y}
| c = {{Defof|Metric Induced by Norm on Division Ring|Metric Induced by $\norm {\,\cdot\,}$}}
}}
{{eqn | r = \norm {\paren {x - z} + \paren {z - y} }
}}
{{eqn | r = \max \set {\norm {x - z}, \norm {z - y} }
| o = \le
... | Let $\struct {R, \norm {\,\cdot\,}}$ be a [[Definition:Non-Archimedean Division Ring Norm|non-Archimedean normed division ring]].
Let $d$ be the [[Definition:Metric Induced by Norm on Division Ring|metric induced by $\norm {\,\cdot\,}$]].
Then $d$ is a [[Definition:Non-Archimedean Metric|non-Archimedean metric]]. | Let $x, y, z \in R$.
{{begin-eqn}}
{{eqn | l = \map d {x, y}
| r = \norm {x - y}
| c = {{Defof|Metric Induced by Norm on Division Ring|Metric Induced by $\norm {\,\cdot\,}$}}
}}
{{eqn | r = \norm {\paren {x - z} + \paren {z - y} }
}}
{{eqn | r = \max \set {\norm {x - z}, \norm {z - y} }
| o = \le
... | Non-Archimedean Norm iff Non-Archimedean Metric/Necessary Condition | https://proofwiki.org/wiki/Non-Archimedean_Norm_iff_Non-Archimedean_Metric/Necessary_Condition | https://proofwiki.org/wiki/Non-Archimedean_Norm_iff_Non-Archimedean_Metric/Necessary_Condition | [
"Normed Division Rings",
"Metric Spaces",
"Non-Archimedean Norms"
] | [
"Definition:Non-Archimedean/Norm (Division Ring)",
"Definition:Metric Induced by Norm on Division Ring",
"Definition:Non-Archimedean/Metric"
] | [] |
proofwiki-14883 | Non-Archimedean Norm iff Non-Archimedean Metric/Sufficient Condition | Let $\struct {R, \norm {\,\cdot\,} }$ be a normed division ring with zero $0$.
Let $d$ be the metric induced by $\norm {\, \cdot \,}$.
Let $d$ be non-Archimedean.
Then:
:$\norm {\, \cdot \,}$ is a non-Archimedean norm. | Let $x, y \in R$.
{{begin-eqn}}
{{eqn | l = \norm {x + y}
| r = \norm {x - \paren {-y} }
}}
{{eqn | r = \map d {x, - y}
| c = {{Defof|Metric Induced by Norm on Division Ring|Metric Induced by $\norm {\, \cdot \,}$}}
}}
{{eqn | r = \max \set {\map d {x, 0}, \map d {0, -y} }
| o = \le
| c = {{Defo... | Let $\struct {R, \norm {\,\cdot\,} }$ be a [[Definition:Normed Division Ring|normed division ring]] with [[Definition:Ring Zero|zero]] $0$.
Let $d$ be the [[Definition:Metric Induced by Norm on Division Ring|metric induced by $\norm {\, \cdot \,}$]].
Let $d$ be [[Definition:Non-Archimedean Metric|non-Archimedean]].
... | Let $x, y \in R$.
{{begin-eqn}}
{{eqn | l = \norm {x + y}
| r = \norm {x - \paren {-y} }
}}
{{eqn | r = \map d {x, - y}
| c = {{Defof|Metric Induced by Norm on Division Ring|Metric Induced by $\norm {\, \cdot \,}$}}
}}
{{eqn | r = \max \set {\map d {x, 0}, \map d {0, -y} }
| o = \le
| c = {{Def... | Non-Archimedean Norm iff Non-Archimedean Metric/Sufficient Condition | https://proofwiki.org/wiki/Non-Archimedean_Norm_iff_Non-Archimedean_Metric/Sufficient_Condition | https://proofwiki.org/wiki/Non-Archimedean_Norm_iff_Non-Archimedean_Metric/Sufficient_Condition | [
"Normed Division Rings",
"Metric Spaces",
"Non-Archimedean Norms"
] | [
"Definition:Normed Division Ring",
"Definition:Ring Zero",
"Definition:Metric Induced by Norm on Division Ring",
"Definition:Non-Archimedean/Metric",
"Definition:Non-Archimedean/Norm (Division Ring)"
] | [] |
proofwiki-14884 | Non-Archimedean Norm iff Non-Archimedean Metric | Let $\struct {R, \norm {\, \cdot \,} }$ be a normed division ring with zero $0$.
Let $d$ be the metric induced by $\norm {\,\cdot\,}$.
Then:
:$\norm {\, \cdot \,}$ is a non-Archimedean norm {{iff}} $d$ is a non-Archimedean metric. | === Necessary Condition ===
{{:Non-Archimedean Norm iff Non-Archimedean Metric/Necessary Condition}}{{qed|lemma}} | Let $\struct {R, \norm {\, \cdot \,} }$ be a [[Definition:Normed Division Ring|normed division ring]] with [[Definition:Ring Zero|zero]] $0$.
Let $d$ be the [[Definition:Metric Induced by Norm on Division Ring|metric induced by $\norm {\,\cdot\,}$]].
Then:
:$\norm {\, \cdot \,}$ is a [[Definition:Non-Archimedean Div... | === [[Non-Archimedean Norm iff Non-Archimedean Metric/Necessary Condition|Necessary Condition]] ===
{{:Non-Archimedean Norm iff Non-Archimedean Metric/Necessary Condition}}{{qed|lemma}} | Non-Archimedean Norm iff Non-Archimedean Metric | https://proofwiki.org/wiki/Non-Archimedean_Norm_iff_Non-Archimedean_Metric | https://proofwiki.org/wiki/Non-Archimedean_Norm_iff_Non-Archimedean_Metric | [
"Normed Division Rings",
"Metric Spaces",
"Non-Archimedean Norms"
] | [
"Definition:Normed Division Ring",
"Definition:Ring Zero",
"Definition:Metric Induced by Norm on Division Ring",
"Definition:Non-Archimedean/Norm (Division Ring)",
"Definition:Non-Archimedean/Metric"
] | [
"Non-Archimedean Norm iff Non-Archimedean Metric/Necessary Condition"
] |
proofwiki-14885 | Quotients of 3 Unequal Numbers are Unequal | Let $x, y, z \in \R_{\ne 0}$ be non-zero real numbers which are not all equal.
Then $\dfrac x y, \dfrac y z, \dfrac z x$ are also not all equal. | {{AimForCont}} $\dfrac x y = \dfrac y z = \dfrac z x$.
{{begin-eqn}}
{{eqn | o =
| r = \dfrac x y = \dfrac y z = \dfrac z x
| c =
}}
{{eqn | ll= \leadsto
| o =
| r = x^2 z = y^2 x = z^2 y
| c = multiplying top and bottom by $x y z$
}}
{{eqn | ll= \leadsto
| o =
| r = x z = ... | Let $x, y, z \in \R_{\ne 0}$ be non-[[Definition:Zero (Number)|zero]] [[Definition:Real Number|real numbers]] which are not all equal.
Then $\dfrac x y, \dfrac y z, \dfrac z x$ are also not all equal. | {{AimForCont}} $\dfrac x y = \dfrac y z = \dfrac z x$.
{{begin-eqn}}
{{eqn | o =
| r = \dfrac x y = \dfrac y z = \dfrac z x
| c =
}}
{{eqn | ll= \leadsto
| o =
| r = x^2 z = y^2 x = z^2 y
| c = multiplying [[Definition:Numerator|top]] and [[Definition:Denominator|bottom]] by $x y z$
}}... | Quotients of 3 Unequal Numbers are Unequal | https://proofwiki.org/wiki/Quotients_of_3_Unequal_Numbers_are_Unequal | https://proofwiki.org/wiki/Quotients_of_3_Unequal_Numbers_are_Unequal | [
"Real Division"
] | [
"Definition:Zero (Number)",
"Definition:Real Number"
] | [
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Definition:Contradiction",
"Proof by Contradiction"
] |
proofwiki-14886 | Union of Power Sets not always Equal to Powerset of Union | The union of the power sets of two sets $S$ and $T$ is not necessarily equal to the power set of their union. | Proof by Counterexample:
Let $S = \set {1, 2, 3}, T = \set {2, 3, 4}, X = \set {1, 2, 3, 4}$.
{{begin-eqn}}
{{eqn | l = S \cup T
| r = \set {1, 2, 3, 4}
| c =
}}
{{eqn | ll= \leadsto
| l = X
| o = \subseteq
| r = S \cup T
| c =
}}
{{eqn | ll= \leadsto
| l = X
| o = \in
... | The [[Definition:Set Union|union]] of the [[Definition:Power Set|power sets]] of two [[Definition:Set|sets]] $S$ and $T$ is not necessarily equal to the [[Definition:Power Set|power set]] of their [[Definition:Set Union|union]]. | [[Proof by Counterexample]]:
Let $S = \set {1, 2, 3}, T = \set {2, 3, 4}, X = \set {1, 2, 3, 4}$.
{{begin-eqn}}
{{eqn | l = S \cup T
| r = \set {1, 2, 3, 4}
| c =
}}
{{eqn | ll= \leadsto
| l = X
| o = \subseteq
| r = S \cup T
| c =
}}
{{eqn | ll= \leadsto
| l = X
| o... | Union of Power Sets not always Equal to Powerset of Union | https://proofwiki.org/wiki/Union_of_Power_Sets_not_always_Equal_to_Powerset_of_Union | https://proofwiki.org/wiki/Union_of_Power_Sets_not_always_Equal_to_Powerset_of_Union | [
"Power Set",
"Set Union",
"Subsets"
] | [
"Definition:Set Union",
"Definition:Power Set",
"Definition:Set",
"Definition:Power Set",
"Definition:Set Union"
] | [
"Proof by Counterexample"
] |
proofwiki-14887 | Limit of Intersection of Closed Intervals from Zero to Positive Integer Reciprocal | For all (strictly) positive integers $n \in \Z_{>0}$, let $A_n$ be the closed real interval:
:$A_n = \closedint 0 {\dfrac 1 n}$
Let $A \subseteq \R$ be the subset of the real numbers defined as:
:$A = \ds \lim_{n \mathop \to \infty} \bigcap A_n$
Then:
:$A = \set 0$ | First it is noted that:
:$\forall x \in \R_{<0}: x \notin A$
and that by definition of closed real interval:
:$\forall n \in \Z_{>0}: 0 \in A_n$
and so by definition of intersection:
:$0 \in A$
It remains to demonstrate that:
:$\forall x \in \R_{>0}: x \notin A$
{{AimForCont}} $\exists x \in \R_{>0}: x \in A$.
By the A... | For all [[Definition:Strictly Positive Integer|(strictly) positive integers]] $n \in \Z_{>0}$, let $A_n$ be the [[Definition:Closed Real Interval|closed real interval]]:
:$A_n = \closedint 0 {\dfrac 1 n}$
Let $A \subseteq \R$ be the [[Definition:Subset|subset]] of the [[Definition:Real Number|real numbers]] defined as... | First it is noted that:
:$\forall x \in \R_{<0}: x \notin A$
and that by definition of [[Definition:Closed Real Interval|closed real interval]]:
:$\forall n \in \Z_{>0}: 0 \in A_n$
and so by definition of [[Definition:Set Intersection|intersection]]:
:$0 \in A$
It remains to demonstrate that:
:$\forall x \in \R_{>0}... | Limit of Intersection of Closed Intervals from Zero to Positive Integer Reciprocal | https://proofwiki.org/wiki/Limit_of_Intersection_of_Closed_Intervals_from_Zero_to_Positive_Integer_Reciprocal | https://proofwiki.org/wiki/Limit_of_Intersection_of_Closed_Intervals_from_Zero_to_Positive_Integer_Reciprocal | [
"Real Analysis"
] | [
"Definition:Strictly Positive/Integer",
"Definition:Real Interval/Closed",
"Definition:Subset",
"Definition:Real Number"
] | [
"Definition:Real Interval/Closed",
"Definition:Set Intersection",
"Axiom of Archimedes",
"Reciprocal Function is Strictly Decreasing",
"Definition:Set Intersection",
"Definition:Contradiction",
"Definition:Element"
] |
proofwiki-14888 | Total Number of Set Partitions | Let $S$ be a finite set of cardinality $n$.
Then the number of different partitions of $S$ is $B_n$, where $B_n$ is the $n$th Bell number. | The number of different partitions of $S$ is '''defined''' as $B_n$.
From Bell Number as Summation over Lower Index of Stirling Numbers of the Second Kind:
:$B_n = \ds \sum_{k \mathop = 0}^n {n \brace k}$
where $\ds {n \brace k}$ denotes a Stirling number of the second kind.
From Number of Set Partitions by Number of C... | Let $S$ be a [[Definition:Finite Set|finite set]] of [[Definition:Cardinality|cardinality]] $n$.
Then the number of different [[Definition:Set Partition|partitions]] of $S$ is $B_n$, where $B_n$ is the $n$th [[Definition:Bell Number|Bell number]]. | The number of different [[Definition:Set Partition|partitions]] of $S$ is '''defined''' as $B_n$.
From [[Bell Number as Summation over Lower Index of Stirling Numbers of the Second Kind]]:
:$B_n = \ds \sum_{k \mathop = 0}^n {n \brace k}$
where $\ds {n \brace k}$ denotes a [[Definition:Stirling Numbers of the Second Ki... | Total Number of Set Partitions | https://proofwiki.org/wiki/Total_Number_of_Set_Partitions | https://proofwiki.org/wiki/Total_Number_of_Set_Partitions | [
"Set Partitions"
] | [
"Definition:Finite Set",
"Definition:Cardinality",
"Definition:Set Partition",
"Definition:Bell Number"
] | [
"Definition:Set Partition",
"Bell Number as Summation over Lower Index of Stirling Numbers of the Second Kind",
"Definition:Stirling Numbers of the Second Kind",
"Number of Set Partitions by Number of Components",
"Definition:Set Partition",
"Definition:Set Partition/Component"
] |
proofwiki-14889 | Combination Theorem for Sequences/Normed Division Ring/Inverse Rule | Suppose $l \ne 0$.
Then:
:$\exists k \in \N : \forall n \in \N: x_{k + n} \ne 0$
and the subsequence $\sequence { x_{k+n}^{-1} }$ is well-defined and convergent with:
:$\ds \lim_{n \mathop \to \infty} {x_{k + n} }^{-1} = l^{-1}$. | Since $\sequence {x_n}$ converges to $l \ne 0$, by Sequence Converges to Within Half Limit then:
:$\exists k \in \N: \forall n \in \N: \dfrac {\norm l} 2 < \norm {x_{k + n} }$
By {{Norm-axiom-mult|1}}:
:$\forall n \in \N : x_{k + n} \ne 0$
Let $\sequence {y_n}$ be the subsequence of $\sequence {x_n}$ where $y_n = x_{k ... | Suppose $l \ne 0$.
Then:
:$\exists k \in \N : \forall n \in \N: x_{k + n} \ne 0$
and the [[Definition:Subsequence|subsequence]] $\sequence { x_{k+n}^{-1} }$ is well-defined and [[Definition:Convergent Sequence in Normed Division Ring|convergent]] with:
:$\ds \lim_{n \mathop \to \infty} {x_{k + n} }^{-1} = l^{-1}$. | Since $\sequence {x_n}$ [[Definition:Convergent Sequence in Normed Division Ring|converges]] to $l \ne 0$, by [[Sequence Converges to Within Half Limit/Normed Division Ring|Sequence Converges to Within Half Limit]] then:
:$\exists k \in \N: \forall n \in \N: \dfrac {\norm l} 2 < \norm {x_{k + n} }$
By {{Norm-axiom-mul... | Combination Theorem for Sequences/Normed Division Ring/Inverse Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Sequences/Normed_Division_Ring/Inverse_Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Sequences/Normed_Division_Ring/Inverse_Rule | [
"Combination Theorem for Sequences in Normed Division Rings"
] | [
"Definition:Subsequence",
"Definition:Convergent Sequence/Normed Division Ring"
] | [
"Definition:Convergent Sequence/Normed Division Ring",
"Sequence Converges to Within Half Limit/Normed Division Ring",
"Definition:Subsequence",
"Limit of Subsequence equals Limit of Sequence/Normed Division Ring",
"Definition:Convergent Sequence/Normed Division Ring",
"Definition:Well-Defined"
] |
proofwiki-14890 | Bell Number as Summation over Lower Index of Stirling Numbers of the Second Kind | Let $B_n$ be the Bell number for $n \in \Z_{\ge 0}$.
Then:
:$B_n = \ds \sum_{k \mathop = 0}^n {n \brace k}$
where $\ds {n \brace k}$ denotes a Stirling number of the second kind. | By definition of Bell numbers:
:$B_n$ is the number of partitions of a (finite) set whose cardinality is $n$.
First consider the case where $n > 0$.
From Number of Set Partitions by Number of Components, the number of partitions of $S$ into $k$ components is $\ds {n \brace k}$.
Thus the total number of all partitions o... | Let $B_n$ be the [[Definition:Bell Number|Bell number]] for $n \in \Z_{\ge 0}$.
Then:
:$B_n = \ds \sum_{k \mathop = 0}^n {n \brace k}$
where $\ds {n \brace k}$ denotes a [[Definition:Stirling Numbers of the Second Kind|Stirling number of the second kind]]. | By definition of [[Definition:Bell Number|Bell numbers]]:
:$B_n$ is the number of [[Definition:Set Partition|partitions]] of a [[Definition:Finite Set|(finite) set]] whose [[Definition:Cardinality|cardinality]] is $n$.
First consider the case where $n > 0$.
From [[Number of Set Partitions by Number of Components]],... | Bell Number as Summation over Lower Index of Stirling Numbers of the Second Kind | https://proofwiki.org/wiki/Bell_Number_as_Summation_over_Lower_Index_of_Stirling_Numbers_of_the_Second_Kind | https://proofwiki.org/wiki/Bell_Number_as_Summation_over_Lower_Index_of_Stirling_Numbers_of_the_Second_Kind | [
"Bell Numbers",
"Stirling Numbers"
] | [
"Definition:Bell Number",
"Definition:Stirling Numbers of the Second Kind"
] | [
"Definition:Bell Number",
"Definition:Set Partition",
"Definition:Finite Set",
"Definition:Cardinality",
"Number of Set Partitions by Number of Components",
"Definition:Set Partition",
"Definition:Set Partition/Component",
"Definition:Set Partition",
"Definition:Set Partition",
"Category:Bell Numb... |
proofwiki-14891 | Bell Number as Summation over Lower Index of Stirling Numbers of the Second Kind/Corollary | :$B_n = \ds \sum_{k \mathop = 1}^n {n \brace k}$ | From Bell Number as Summation over Lower Index of Stirling Numbers of the Second Kind, we have that:
:$B_n = \ds \sum_{k \mathop = 0}^n {n \brace k}$
But when $n > 0$:
:$\ds {n \brace 0} = 0$
Hence the result.
{{qed}}
Category:Bell Numbers
Category:Stirling Numbers
tneu45yoadsobh8o6qajwn77r3ju1la | :$B_n = \ds \sum_{k \mathop = 1}^n {n \brace k}$ | From [[Bell Number as Summation over Lower Index of Stirling Numbers of the Second Kind]], we have that:
:$B_n = \ds \sum_{k \mathop = 0}^n {n \brace k}$
But when $n > 0$:
:$\ds {n \brace 0} = 0$
Hence the result.
{{qed}}
[[Category:Bell Numbers]]
[[Category:Stirling Numbers]]
tneu45yoadsobh8o6qajwn77r3ju1la | Bell Number as Summation over Lower Index of Stirling Numbers of the Second Kind/Corollary | https://proofwiki.org/wiki/Bell_Number_as_Summation_over_Lower_Index_of_Stirling_Numbers_of_the_Second_Kind/Corollary | https://proofwiki.org/wiki/Bell_Number_as_Summation_over_Lower_Index_of_Stirling_Numbers_of_the_Second_Kind/Corollary | [
"Bell Numbers",
"Stirling Numbers"
] | [] | [
"Bell Number as Summation over Lower Index of Stirling Numbers of the Second Kind",
"Category:Bell Numbers",
"Category:Stirling Numbers"
] |
proofwiki-14892 | Equivalence of Well-Ordering Principle and Induction/Proof/PFI implies PCI | The Principle of Finite Induction implies the Principle of Complete Finite Induction.
That is:
:Principle of Finite Induction: Given a subset $S \subseteq \N$ of the natural numbers which has these properties:
::$0 \in S$
::$n \in S \implies n + 1 \in S$
:then $S = \N$.
implies:
:Principle of Complete Finite Induction:... | Let us assume that the '''PFI''' is true.
Let $S \subseteq \N$ which satisfy:
:$(A): \quad 0 \in S$
:$(B): \quad \set {0, 1, \ldots, n} \subseteq S \implies n + 1 \in S$.
We want to show that $S = \N$, that is, the '''PCI''' is true.
Let $P \paren n$ be the propositional function:
:$P \paren n \iff \set {0, 1, \ldots, ... | The [[Principle of Finite Induction]] implies the [[Principle of Complete Finite Induction]].
That is:
:[[Principle of Finite Induction]]: Given a [[Definition:Subset|subset]] $S \subseteq \N$ of the [[Definition:Natural Numbers|natural numbers]] which has these properties:
::$0 \in S$
::$n \in S \implies n + 1 \in ... | Let us assume that the '''[[Principle of Finite Induction|PFI]]''' is true.
Let $S \subseteq \N$ which satisfy:
:$(A): \quad 0 \in S$
:$(B): \quad \set {0, 1, \ldots, n} \subseteq S \implies n + 1 \in S$.
We want to show that $S = \N$, that is, the '''[[Principle of Complete Finite Induction|PCI]]''' is true.
Let $... | Equivalence of Well-Ordering Principle and Induction/Proof/PFI implies PCI | https://proofwiki.org/wiki/Equivalence_of_Well-Ordering_Principle_and_Induction/Proof/PFI_implies_PCI | https://proofwiki.org/wiki/Equivalence_of_Well-Ordering_Principle_and_Induction/Proof/PFI_implies_PCI | [
"Equivalence of Well-Ordering Principle and Induction"
] | [
"Principle of Finite Induction",
"Second Principle of Finite Induction",
"Principle of Finite Induction",
"Definition:Subset",
"Definition:Natural Numbers",
"Second Principle of Finite Induction",
"Definition:Subset",
"Definition:Natural Numbers"
] | [
"Principle of Finite Induction",
"Second Principle of Finite Induction",
"Definition:Propositional Function",
"Principle of Finite Induction",
"Principle of Finite Induction",
"Second Principle of Finite Induction"
] |
proofwiki-14893 | Equivalence of Well-Ordering Principle and Induction/Proof/PCI implies WOP | The Principle of Complete Induction implies the Well-Ordering Principle.
That is:
:Principle of Complete Induction: Given a subset $S \subseteq \N$ of the natural numbers which has these properties:
::$0 \in S$
::$\set {0, 1, \ldots, n} \subseteq S \implies n + 1 \in S$
:then $S = \N$.
implies:
:Well-Ordering Principle... | Let us assume that the '''PCI''' is true.
Let $\O \subsetneqq S \subseteq \N$.
We need to show that $S$ has a minimal element, and so demonstrate that the '''WOP''' holds.
{{AimForCont}} that:
:$(C): \quad S$ has no minimal element.
Let $\map P n$ be the propositional function:
:$n \notin S$
Suppose $0 \in S$.
We have ... | The [[Principle of Complete Induction]] implies the [[Well-Ordering Principle]].
That is:
:[[Principle of Complete Induction]]: Given a [[Definition:Subset|subset]] $S \subseteq \N$ of the [[Definition:Natural Numbers|natural numbers]] which has these properties:
::$0 \in S$
::$\set {0, 1, \ldots, n} \subseteq S \im... | Let us assume that the '''[[Principle of Complete Induction|PCI]]''' is true.
Let $\O \subsetneqq S \subseteq \N$.
We need to show that $S$ has a [[Definition:Minimal Element|minimal element]], and so demonstrate that the '''[[Well-Ordering Principle|WOP]]''' holds.
{{AimForCont}} that:
:$(C): \quad S$ has no [[Defi... | Equivalence of Well-Ordering Principle and Induction/Proof/PCI implies WOP | https://proofwiki.org/wiki/Equivalence_of_Well-Ordering_Principle_and_Induction/Proof/PCI_implies_WOP | https://proofwiki.org/wiki/Equivalence_of_Well-Ordering_Principle_and_Induction/Proof/PCI_implies_WOP | [
"Equivalence of Well-Ordering Principle and Induction"
] | [
"Second Principle of Mathematical Induction",
"Well-Ordering Principle",
"Second Principle of Mathematical Induction",
"Definition:Subset",
"Definition:Natural Numbers",
"Well-Ordering Principle",
"Definition:Non-Empty Set",
"Definition:Subset",
"Definition:Minimal/Element"
] | [
"Second Principle of Mathematical Induction",
"Definition:Minimal/Element",
"Well-Ordering Principle",
"Definition:Minimal/Element",
"Definition:Propositional Function",
"Definition:Lower Bound of Set",
"Lower Bound for Subset",
"Definition:Lower Bound of Set",
"Definition:Minimal/Element",
"Defin... |
proofwiki-14894 | Equivalence of Well-Ordering Principle and Induction/Proof/WOP implies PFI | The Well-Ordering Principle implies the Principle of Finite Induction.
That is:
:Well-Ordering Principle: Every non-empty subset of $\N$ has a minimal element
implies:
:Principle of Finite Induction: Given a subset $S \subseteq \N$ of the natural numbers which has these properties:
::$0 \in S$
::$n \in S \implies n + 1... | We assume the truth of '''WOP'''.
Let $S \subseteq \N$ which satisfy:
:$(D): \quad 0 \in S$
:$(E): \quad n \in S \implies n+1 \in S$.
We want to show that $S = \N$, that is, the '''PFI''' is true.
{{AimForCont}} that:
:$S \ne \N$
Consider $S' = \N \setminus S$, where $\setminus$ denotes set difference.
From Set Differe... | The [[Well-Ordering Principle]] implies the [[Principle of Finite Induction]].
That is:
:[[Well-Ordering Principle]]: Every [[Definition:Non-Empty Set|non-empty]] [[Definition:Subset|subset]] of $\N$ has a [[Definition:Minimal Element|minimal element]]
implies:
:[[Principle of Finite Induction]]: Given a [[Definit... | We assume the truth of '''[[Well-Ordering Principle|WOP]]'''.
Let $S \subseteq \N$ which satisfy:
:$(D): \quad 0 \in S$
:$(E): \quad n \in S \implies n+1 \in S$.
We want to show that $S = \N$, that is, the '''[[Principle of Finite Induction|PFI]]''' is true.
{{AimForCont}} that:
:$S \ne \N$
Consider $S' = \N \setm... | Equivalence of Well-Ordering Principle and Induction/Proof/WOP implies PFI | https://proofwiki.org/wiki/Equivalence_of_Well-Ordering_Principle_and_Induction/Proof/WOP_implies_PFI | https://proofwiki.org/wiki/Equivalence_of_Well-Ordering_Principle_and_Induction/Proof/WOP_implies_PFI | [
"Equivalence of Well-Ordering Principle and Induction"
] | [
"Well-Ordering Principle",
"Principle of Finite Induction",
"Well-Ordering Principle",
"Definition:Non-Empty Set",
"Definition:Subset",
"Definition:Minimal/Element",
"Principle of Finite Induction",
"Definition:Subset",
"Definition:Natural Numbers"
] | [
"Well-Ordering Principle",
"Principle of Finite Induction",
"Definition:Set Difference",
"Set Difference is Subset",
"Well-Ordering Principle",
"Definition:Minimal/Element",
"Definition:Lower Bound of Set",
"Lower Bound for Subset",
"Definition:Lower Bound of Set",
"Definition:Set Difference",
"... |
proofwiki-14895 | Floor Function/Examples/Floor of 5 over 2 | :$\floor {\dfrac 5 2} = 2$ | We have that:
{{begin-eqn}}
{{eqn | l = \dfrac 5 2
| r = 2 + \dfrac 1 2
| c =
}}
{{eqn | ll= \leadsto
| l = 2
| o = \le
| r = \dfrac 5 2
| c =
}}
{{eqn | o = <
| r = 3
| c =
}}
{{end-eqn}}
Hence $2$ is the floor of $\dfrac 5 2$ by definition.
{{qed}} | :$\floor {\dfrac 5 2} = 2$ | We have that:
{{begin-eqn}}
{{eqn | l = \dfrac 5 2
| r = 2 + \dfrac 1 2
| c =
}}
{{eqn | ll= \leadsto
| l = 2
| o = \le
| r = \dfrac 5 2
| c =
}}
{{eqn | o = <
| r = 3
| c =
}}
{{end-eqn}}
Hence $2$ is the [[Definition:Floor Function|floor]] of $\dfrac 5 2$ by defin... | Floor Function/Examples/Floor of 5 over 2 | https://proofwiki.org/wiki/Floor_Function/Examples/Floor_of_5_over_2 | https://proofwiki.org/wiki/Floor_Function/Examples/Floor_of_5_over_2 | [
"Examples of Floor Function"
] | [] | [
"Definition:Floor Function"
] |
proofwiki-14896 | Floor Function/Examples/Floor of Minus 5 over 2 | :$\floor {-\dfrac 5 2} = -3$ | We have that:
{{begin-eqn}}
{{eqn | l = -\dfrac 5 2
| r = -3 + \dfrac 1 2
| c =
}}
{{eqn | ll= \leadsto
| l = -3
| o = \le
| r = -\dfrac 5 2
| c =
}}
{{eqn | o = <
| r = -2
| c =
}}
{{end-eqn}}
Hence $-3$ is the floor of $-\dfrac 5 2$ by definition.
{{qed}} | :$\floor {-\dfrac 5 2} = -3$ | We have that:
{{begin-eqn}}
{{eqn | l = -\dfrac 5 2
| r = -3 + \dfrac 1 2
| c =
}}
{{eqn | ll= \leadsto
| l = -3
| o = \le
| r = -\dfrac 5 2
| c =
}}
{{eqn | o = <
| r = -2
| c =
}}
{{end-eqn}}
Hence $-3$ is the [[Definition:Floor Function|floor]] of $-\dfrac 5 2$ b... | Floor Function/Examples/Floor of Minus 5 over 2 | https://proofwiki.org/wiki/Floor_Function/Examples/Floor_of_Minus_5_over_2 | https://proofwiki.org/wiki/Floor_Function/Examples/Floor_of_Minus_5_over_2 | [
"Examples of Floor Function"
] | [] | [
"Definition:Floor Function"
] |
proofwiki-14897 | Floor Function/Examples/Floor of 14 | :$\floor {14} = 14$ | We have that $14$ is an integer.
Thus this is a specific example of Real Number is Integer iff equals Floor:
$\floor x = x \iff x \in \Z$
{{qed}} | :$\floor {14} = 14$ | We have that $14$ is an [[Definition:Integer|integer]].
Thus this is a specific example of [[Real Number is Integer iff equals Floor]]:
$\floor x = x \iff x \in \Z$
{{qed}} | Floor Function/Examples/Floor of 14 | https://proofwiki.org/wiki/Floor_Function/Examples/Floor_of_14 | https://proofwiki.org/wiki/Floor_Function/Examples/Floor_of_14 | [
"Examples of Floor Function"
] | [] | [
"Definition:Integer",
"Real Number is Integer iff equals Floor"
] |
proofwiki-14898 | Floor Function/Examples/Floor of Root 10 | :$\floor {\sqrt {10} } = 3$ | {{begin-eqn}}
{{eqn | l = \sqrt 9
| o = \le
| r = \sqrt 10
| c =
}}
{{eqn | o = <
| r = \sqrt 16
| c =
}}
{{eqn | ll= \leadsto
| l = 3
| o = \le
| r = \sqrt 10
| c =
}}
{{eqn | o = <
| r = 4
| c =
}}
{{end-eqn}}
Hence $3$ is the floor of $\sqrt {10}$... | :$\floor {\sqrt {10} } = 3$ | {{begin-eqn}}
{{eqn | l = \sqrt 9
| o = \le
| r = \sqrt 10
| c =
}}
{{eqn | o = <
| r = \sqrt 16
| c =
}}
{{eqn | ll= \leadsto
| l = 3
| o = \le
| r = \sqrt 10
| c =
}}
{{eqn | o = <
| r = 4
| c =
}}
{{end-eqn}}
Hence $3$ is the [[Definition:Floor F... | Floor Function/Examples/Floor of Root 10 | https://proofwiki.org/wiki/Floor_Function/Examples/Floor_of_Root_10 | https://proofwiki.org/wiki/Floor_Function/Examples/Floor_of_Root_10 | [
"Examples of Floor Function"
] | [] | [
"Definition:Floor Function"
] |
proofwiki-14899 | Combination Theorem for Cauchy Sequences/Inverse Rule | Suppose $\sequence {x_n}$ does not converge to $0$.
Then:
:$\exists K \in \N: \forall n > K : x_n \ne 0$
and the sequence:
:$\sequence {\paren {x_{K + n} }^{-1} }_{n \mathop \in \N}$ is well-defined and a Cauchy sequence. | Since $\sequence {x_n}$ does not converge to $0$, by Cauchy Sequence Is Eventually Bounded Away From Non-Limit then:
:$\exists K \in \N$ and $C \in \R_{>0}: \forall n > K: C < \norm {x_n}$
or equivalently:
:$\exists K \in \N$ and $C \in \R_{>0}: \forall n > K: 1 < \dfrac {\norm {x_n} } C$
By {{Norm-axiom-mult|1}}:
:$\... | Suppose $\sequence {x_n}$ does not [[Definition:Convergent Sequence in Normed Division Ring|converge]] to $0$.
Then:
:$\exists K \in \N: \forall n > K : x_n \ne 0$
and the [[Definition:Sequence|sequence]]:
:$\sequence {\paren {x_{K + n} }^{-1} }_{n \mathop \in \N}$ is well-defined and a [[Definition:Cauchy Sequence in... | Since $\sequence {x_n}$ does not [[Definition:Convergent Sequence in Normed Division Ring|converge]] to $0$, by [[Cauchy Sequence Is Eventually Bounded Away From Non-Limit]] then:
:$\exists K \in \N$ and $C \in \R_{>0}: \forall n > K: C < \norm {x_n}$
or equivalently:
:$\exists K \in \N$ and $C \in \R_{>0}: \forall n ... | Combination Theorem for Cauchy Sequences/Inverse Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Cauchy_Sequences/Inverse_Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Cauchy_Sequences/Inverse_Rule | [
"Combination Theorem for Cauchy Sequences"
] | [
"Definition:Convergent Sequence/Normed Division Ring",
"Definition:Sequence",
"Definition:Cauchy Sequence/Normed Division Ring"
] | [
"Definition:Convergent Sequence/Normed Division Ring",
"Cauchy Sequence Is Eventually Bounded Away From Non-Limit",
"Definition:Subsequence",
"Subsequence of Cauchy Sequence in Normed Division Ring is Cauchy Sequence",
"Definition:Cauchy Sequence/Normed Division Ring",
"Definition:Cauchy Sequence/Normed D... |
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