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-1500 | Comparison Test | Let $\ds \sum_{n \mathop = 1}^\infty b_n$ be a convergent series of positive real numbers.
Let $\sequence {a_n}$ be a sequence $\R$ or sequence in $\C$.
Let $\forall n \in \N_{>0}: \cmod {a_n} \le b_n$.
Then the series $\ds \sum_{n \mathop = 1}^\infty a_n$ converges absolutely. | Let $\epsilon > 0$.
As $\ds \sum_{n \mathop = 1}^\infty b_n$ converges, its tail tends to zero.
So:
:$\ds \exists N: \forall n > N: \sum_{k \mathop = n + 1}^\infty b_k < \epsilon$
Let $\sequence {s_n}$ be the sequence of partial sums of $\ds \sum_{n \mathop = 1}^\infty a_n$.
Then $\forall n > m > N$:
{{begin-eqn}}
{{eq... | Let $\ds \sum_{n \mathop = 1}^\infty b_n$ be a [[Definition:Convergent Series|convergent series]] of [[Definition:Positive Real Number|positive real numbers]].
Let $\sequence {a_n}$ be a [[Definition:Real Sequence|sequence $\R$]] or [[Definition:Complex Sequence|sequence in $\C$]].
Let $\forall n \in \N_{>0}: \cmod ... | Let $\epsilon > 0$.
As $\ds \sum_{n \mathop = 1}^\infty b_n$ [[Definition:Convergent Series|converges]], its [[Tail of Convergent Series tends to Zero|tail tends to zero]].
So:
:$\ds \exists N: \forall n > N: \sum_{k \mathop = n + 1}^\infty b_k < \epsilon$
Let $\sequence {s_n}$ be the [[Definition:Series|sequence o... | Comparison Test | https://proofwiki.org/wiki/Comparison_Test | https://proofwiki.org/wiki/Comparison_Test | [
"Comparison Test",
"Series",
"Convergence Tests",
"Named Theorems"
] | [
"Definition:Convergent Series",
"Definition:Positive/Real Number",
"Definition:Real Sequence",
"Definition:Complex Sequence",
"Definition:Series",
"Definition:Absolutely Convergent Series"
] | [
"Definition:Convergent Series",
"Tail of Convergent Series tends to Zero",
"Definition:Series",
"Indexed Summation over Adjacent Intervals",
"Triangle Inequality for Indexed Summations",
"Definition:Cauchy Sequence",
"Real Number Line is Complete Metric Space",
"Complex Plane is Complete Metric Space"... |
proofwiki-1501 | Ratio Test | Let $\ds \sum_{n \mathop = 1}^\infty a_n$ be a series of real numbers in $\R$, or a series of complex numbers in $\C$.
Let the sequence $\sequence {a_n}$ satisfy:
:$\ds \lim_{n \mathop \to \infty} \size {\frac {a_{n + 1} } {a_n} } = l$
:If $l > 1 $, then $\ds \sum_{n \mathop = 1}^\infty a_n$ diverges.
:If $l < 1 $, the... | From the statement of the theorem, it is necessary that $\forall n: a_n \ne 0$; otherwise $\size {\dfrac {a_{n + 1} } {a_n} }$ is not defined.
Here, $\size {\dfrac {a_{n + 1} } {a_n} }$ denotes either the absolute value of $\dfrac {a_{n + 1} } {a_n}$, or the complex modulus of $\dfrac {a_{n + 1} } {a_n}$. | Let $\ds \sum_{n \mathop = 1}^\infty a_n$ be a [[Definition:Series|series]] of [[Definition:Real Number|real numbers]] in $\R$, or a [[Definition:Series|series]] of [[Definition:Complex Number|complex numbers]] in $\C$.
Let the [[Definition:Sequence|sequence]] $\sequence {a_n}$ satisfy:
:$\ds \lim_{n \mathop \to \inft... | From the statement of the theorem, it is necessary that $\forall n: a_n \ne 0$; otherwise $\size {\dfrac {a_{n + 1} } {a_n} }$ is not defined.
Here, $\size {\dfrac {a_{n + 1} } {a_n} }$ denotes either the [[Definition:Absolute Value|absolute value]] of $\dfrac {a_{n + 1} } {a_n}$, or the [[Definition:Complex Modulus|c... | Ratio Test | https://proofwiki.org/wiki/Ratio_Test | https://proofwiki.org/wiki/Ratio_Test | [
"Ratio Test",
"Convergence Tests",
"Named Theorems"
] | [
"Definition:Series",
"Definition:Real Number",
"Definition:Series",
"Definition:Complex Number",
"Definition:Sequence",
"Definition:Divergent Series",
"Definition:Absolutely Convergent Series"
] | [
"Definition:Absolute Value",
"Definition:Complex Modulus"
] |
proofwiki-1502 | Limit of Subsequence of Bounded Sequence | Let $\sequence {x_n}$ be a sequence in $\R$.
Let $\sequence {x_n}$ be bounded.
Let $b \in \R$ be a real number.
Suppose that $\forall N: \exists n > N: x_n \ge b$.
Then $\sequence {x_n}$ has a subsequence which converges to a limit $l \ge b$. | Let us pick $N \in \N$.
Then $\exists n_1 > N: x_{n_1} \ge b$.
Again, $\exists n_2 > n_1: x_{n_2} \ge b$.
And so on: for each $n_k$ we find, $\exists n_{k+1} > n_k: x_{n_{k+1}} \ge b$.
In this way we can build a subsequence of $\sequence {x_n}$ each of whose terms are $b$ or bigger.
By the Bolzano-Weierstrass Theorem, ... | Let $\sequence {x_n}$ be a [[Definition:Real Sequence|sequence in $\R$]].
Let $\sequence {x_n}$ be [[Definition:Bounded Real Sequence|bounded]].
Let $b \in \R$ be a [[Definition:Real Number|real number]].
Suppose that $\forall N: \exists n > N: x_n \ge b$.
Then $\sequence {x_n}$ has a [[Definition:Subsequence|subs... | Let us pick $N \in \N$.
Then $\exists n_1 > N: x_{n_1} \ge b$.
Again, $\exists n_2 > n_1: x_{n_2} \ge b$.
And so on: for each $n_k$ we find, $\exists n_{k+1} > n_k: x_{n_{k+1}} \ge b$.
In this way we can build a [[Definition:Subsequence|subsequence]] of $\sequence {x_n}$ each of whose terms are $b$ or bigger.
By t... | Limit of Subsequence of Bounded Sequence | https://proofwiki.org/wiki/Limit_of_Subsequence_of_Bounded_Sequence | https://proofwiki.org/wiki/Limit_of_Subsequence_of_Bounded_Sequence | [
"Limits of Sequences"
] | [
"Definition:Real Sequence",
"Definition:Bounded Sequence/Real",
"Definition:Real Number",
"Definition:Subsequence",
"Definition:Convergent Sequence/Real Numbers",
"Definition:Limit of Sequence/Real Numbers"
] | [
"Definition:Subsequence",
"Bolzano-Weierstrass Theorem",
"Definition:Subsequence",
"Definition:Subsequence",
"Definition:Convergent Sequence/Real Numbers",
"Lower and Upper Bounds for Sequences"
] |
proofwiki-1503 | Terms of Bounded Sequence Within Bounds | Let $\sequence {x_n}$ be a sequence in $\R$.
Let $\sequence {x_n}$ be bounded.
Let the limit superior of $\sequence {x_n}$ be $\overline l$.
Let the limit inferior of $\sequence {x_n}$ be $\underline l$.
Then:
:$\forall \epsilon > 0: \exists N: \forall n > N: x_n < \overline l + \epsilon$
:$\forall \epsilon > 0: \exist... | === Upper Bound ===
First we show that:
:$\forall \epsilon > 0: \exists N: \forall n > N: x_n < \overline l + \epsilon$
{{AimForCont}} this proposition were to be false.
That would mean that for some $\epsilon > 0$ it would be true that for each $N$ we would be able to find $n > N$ such that $x_n \ge \overline l + \eps... | Let $\sequence {x_n}$ be a [[Definition:Real Sequence|sequence in $\R$]].
Let $\sequence {x_n}$ be [[Definition:Bounded Sequence|bounded]].
Let the [[Definition:Limit Superior|limit superior]] of $\sequence {x_n}$ be $\overline l$.
Let the [[Definition:Limit Inferior|limit inferior]] of $\sequence {x_n}$ be $\underl... | === Upper Bound ===
First we show that:
:$\forall \epsilon > 0: \exists N: \forall n > N: x_n < \overline l + \epsilon$
{{AimForCont}} this [[Definition:Proposition|proposition]] were to be [[Definition:False|false]].
That would mean that for some $\epsilon > 0$ it would be true that for each $N$ we would be able t... | Terms of Bounded Sequence Within Bounds | https://proofwiki.org/wiki/Terms_of_Bounded_Sequence_Within_Bounds | https://proofwiki.org/wiki/Terms_of_Bounded_Sequence_Within_Bounds | [
"Limits of Sequences"
] | [
"Definition:Real Sequence",
"Definition:Bounded Sequence",
"Definition:Limit Superior",
"Definition:Limit Inferior"
] | [
"Definition:Proposition",
"Definition:False",
"Limit of Subsequence of Bounded Sequence",
"Definition:Convergent Sequence/Real Numbers",
"Definition:Subsequence",
"Definition:Proposition",
"Definition:False",
"Limit of Subsequence of Bounded Sequence",
"Definition:Convergent Sequence/Real Numbers",
... |
proofwiki-1504 | Convergence of Limsup and Liminf | Let $\sequence {x_n}$ be a sequence in $\R$.
Let the limit superior of $\sequence {x_n}$ be $\overline l$.
Let the limit inferior of $\sequence {x_n}$ be $\underline l$.
Then $\sequence {x_n}$ converges to a limit $l$ {{iff}} $\overline l = \underline l = l$.
Hence a bounded real sequence converges {{iff}} all its conv... | === Sufficient Condition ===
First, suppose that $\overline l = \underline l = l$.
Let $\epsilon > 0$.
By Terms of Bounded Sequence Within Bounds:
:$\exists N_1: \forall n > N_1: x_n < l + \epsilon$
Similarly:
:$\exists N_2: \forall n > N_2: x_n > l - \epsilon$
So take $N = \max \set {N_1, N_2}$.
If $n > N$, both the a... | Let $\sequence {x_n}$ be a [[Definition:Real Sequence|sequence in $\R$]].
Let the [[Definition:Limit Superior|limit superior]] of $\sequence {x_n}$ be $\overline l$.
Let the [[Definition:Limit Inferior|limit inferior]] of $\sequence {x_n}$ be $\underline l$.
Then $\sequence {x_n}$ [[Definition:Convergent Real Seque... | === Sufficient Condition ===
First, suppose that $\overline l = \underline l = l$.
Let $\epsilon > 0$.
By [[Terms of Bounded Sequence Within Bounds]]:
:$\exists N_1: \forall n > N_1: x_n < l + \epsilon$
Similarly:
:$\exists N_2: \forall n > N_2: x_n > l - \epsilon$
So take $N = \max \set {N_1, N_2}$.
If $n > N$, ... | Convergence of Limsup and Liminf | https://proofwiki.org/wiki/Convergence_of_Limsup_and_Liminf | https://proofwiki.org/wiki/Convergence_of_Limsup_and_Liminf | [
"Limits of Sequences",
"Convergence"
] | [
"Definition:Real Sequence",
"Definition:Limit Superior",
"Definition:Limit Inferior",
"Definition:Convergent Sequence/Real Numbers",
"Definition:Limit of Sequence/Real Numbers",
"Definition:Bounded Sequence/Real",
"Definition:Convergent Sequence/Real Numbers",
"Definition:Convergent Sequence/Real Numb... | [
"Terms of Bounded Sequence Within Bounds",
"Negative of Absolute Value"
] |
proofwiki-1505 | Limsup and Liminf are Limits of Bounds | Let $\sequence {x_n}$ be a sequence in $\R$.
Let $\sequence {x_n}$ be bounded.
Let $\ds \overline l = \limsup_{n \mathop \to \infty} x_n$ be the limit superior and $\ds \liminf_{n \mathop \to \infty} x_n$ the limit inferior of $\sequence {x_n}$.
Then:
:$\ds \overline l = \limsup_{n \mathop \to \infty} x_n = \map {\lim_... | First we show that:
:$\ds \limsup_{n \mathop \to \infty} x_n = \map {\lim_{n \mathop \to \infty} } {\sup_{k \mathop \ge n} x_k}$
Let $M_n = \ds \sup_{k \mathop \ge n} x_k$.
By Supremum of Subset, the sequence $\sequence {M_n}$ decreases, for:
:$m \ge n \implies \set {k \in \N: k \ge m} \subseteq \set {k \in \N: k \ge n... | Let $\sequence {x_n}$ be a [[Definition:Real Sequence|sequence in $\R$]].
Let $\sequence {x_n}$ be [[Definition:Bounded Real Sequence|bounded]].
Let $\ds \overline l = \limsup_{n \mathop \to \infty} x_n$ be the [[Definition:Limit Superior|limit superior]] and $\ds \liminf_{n \mathop \to \infty} x_n$ the [[Definition:... | First we show that:
:$\ds \limsup_{n \mathop \to \infty} x_n = \map {\lim_{n \mathop \to \infty} } {\sup_{k \mathop \ge n} x_k}$
Let $M_n = \ds \sup_{k \mathop \ge n} x_k$.
By [[Supremum of Subset]], the [[Definition:Real Sequence|sequence]] $\sequence {M_n}$ [[Definition:Decreasing Real Sequence|decreases]], for:
... | Limsup and Liminf are Limits of Bounds | https://proofwiki.org/wiki/Limsup_and_Liminf_are_Limits_of_Bounds | https://proofwiki.org/wiki/Limsup_and_Liminf_are_Limits_of_Bounds | [
"Limits of Sequences"
] | [
"Definition:Real Sequence",
"Definition:Bounded Sequence/Real",
"Definition:Limit Superior",
"Definition:Limit Inferior"
] | [
"Supremum of Subset",
"Definition:Real Sequence",
"Definition:Decreasing/Sequence/Real Sequence",
"Definition:Lower Bound of Set",
"Definition:Lower Bound of Set",
"Monotone Convergence Theorem (Real Analysis)",
"Definition:Convergent Sequence",
"Bolzano-Weierstrass Theorem",
"Definition:Convergent ... |
proofwiki-1506 | Nth Root Test | Let $\ds \sum_{n \mathop = 1}^\infty a_n$ be a series of real numbers $\R$ or complex numbers $\C$.
Let the sequence $\sequence {a_n}$ be such that the limit superior $\ds \limsup_{n \mathop \to \infty} \size {a_n}^{1/n} = l$.
Then:
:If $l > 1$, the series $\ds \sum_{n \mathop = 1}^\infty a_n$ diverges.
:If $l < 1$, th... | === Absolute Convergence ===
Let $l < 1$.
Then let us choose $\epsilon > 0$ such that $l + \epsilon < 1$.
Consider the real sequence $\sequence {b_n}$ defined by $\sequence {b_n} = \sequence {\size {a_n} }$.
Here, $\size {a_n}$ denotes either the absolute value of $a_n$, or the complex modulus of $a_n$.
Then:
:$\ds l =... | Let $\ds \sum_{n \mathop = 1}^\infty a_n$ be a [[Definition:Series|series]] of [[Definition:Real Number|real numbers]] $\R$ or [[Definition:Complex Number|complex numbers]] $\C$.
Let the [[Definition:Sequence|sequence]] $\sequence {a_n}$ be such that the [[Definition:Limit Superior|limit superior]] $\ds \limsup_{n \ma... | === Absolute Convergence ===
Let $l < 1$.
Then let us choose $\epsilon > 0$ such that $l + \epsilon < 1$.
Consider the [[Definition:Real Number|real]] sequence $\sequence {b_n}$ defined by $\sequence {b_n} = \sequence {\size {a_n} }$.
Here, $\size {a_n}$ denotes either the [[Definition:Absolute Value|absolute value... | Nth Root Test | https://proofwiki.org/wiki/Nth_Root_Test | https://proofwiki.org/wiki/Nth_Root_Test | [
"Nth Root Test",
"Convergence Tests",
"Named Theorems"
] | [
"Definition:Series",
"Definition:Real Number",
"Definition:Complex Number",
"Definition:Sequence",
"Definition:Limit Superior",
"Definition:Divergent Series",
"Definition:Absolutely Convergent Series"
] | [
"Definition:Real Number",
"Definition:Absolute Value",
"Definition:Complex Modulus",
"Terms of Bounded Sequence Within Bounds",
"Sum of Infinite Geometric Sequence",
"Definition:Series",
"Definition:Convergent Sequence",
"Comparison Test",
"Definition:Convergent Sequence",
"Definition:Absolutely C... |
proofwiki-1507 | Series of Power over Factorial Converges | The series $\ds \sum_{n \mathop = 0}^\infty \frac {x^n} {n!}$ converges for all real values of $x$. | If $x = 0$ the result is trivially true as:
:$\forall n \ge 1: \dfrac {0^n} {n!} = 0$
If $x \ne 0$ we have:
:$\size {\dfrac {\paren {\dfrac {x^{n + 1} } {\paren {n + 1}!} } } {\paren {\dfrac {x^n} {n!} } } } = \dfrac {\size x} {n + 1} \to 0$
as $n \to \infty$.
This follows from the results:
:Sequence of Powers of Recip... | The [[Definition:Series|series]] $\ds \sum_{n \mathop = 0}^\infty \frac {x^n} {n!}$ [[Definition:Convergent Series|converges]] for all [[Definition:Real Number|real]] values of $x$. | If $x = 0$ the result is trivially true as:
:$\forall n \ge 1: \dfrac {0^n} {n!} = 0$
If $x \ne 0$ we have:
:$\size {\dfrac {\paren {\dfrac {x^{n + 1} } {\paren {n + 1}!} } } {\paren {\dfrac {x^n} {n!} } } } = \dfrac {\size x} {n + 1} \to 0$
as $n \to \infty$.
This follows from the results:
:[[Sequence of Powers of R... | Series of Power over Factorial Converges | https://proofwiki.org/wiki/Series_of_Power_over_Factorial_Converges | https://proofwiki.org/wiki/Series_of_Power_over_Factorial_Converges | [
"Series"
] | [
"Definition:Series",
"Definition:Convergent Series",
"Definition:Real Number"
] | [
"Sequence of Powers of Reciprocals is Null Sequence",
"Squeeze Theorem/Sequences/Real Numbers",
"Combination Theorem for Sequences/Real/Multiple Rule",
"Ratio Test",
"Definition:Convergent Series",
"Comparison Test",
"Radius of Convergence of Power Series over Factorial"
] |
proofwiki-1508 | Area of Triangle in Terms of Side and Altitude | The area of a triangle $\triangle ABC$ is given by:
:$\dfrac {c \cdot h_c} 2 = \dfrac {b \cdot h_b} 2 = \dfrac {a \cdot h_a} 2$
where:
:$a, b, c$ are the sides
:$h_a, h_b, h_c$ are the altitudes from $A$, $B$ and $C$ respectively. | :400px
Construct a point $D$ so that $\Box ABDC$ is a parallelogram.
From Opposite Sides and Angles of Parallelogram are Equal:
:$\triangle ABC \cong \triangle DCB$
hence their areas are equal.
The Area of Parallelogram is equal to the product of one of its bases and the associated altitude.
Thus
{{begin-eqn}}
{{eqn |... | The [[Definition:Area|area]] of a [[Definition:Triangle (Geometry)|triangle]] $\triangle ABC$ is given by:
:$\dfrac {c \cdot h_c} 2 = \dfrac {b \cdot h_b} 2 = \dfrac {a \cdot h_a} 2$
where:
:$a, b, c$ are the [[Definition:Side of Polygon|sides]]
:$h_a, h_b, h_c$ are the [[Definition:Altitude of Triangle|altitudes]] fro... | :[[File:Area-of-Triangle.png|400px]]
Construct a point $D$ so that $\Box ABDC$ is a [[Definition:Parallelogram|parallelogram]].
From [[Opposite Sides and Angles of Parallelogram are Equal]]:
:$\triangle ABC \cong \triangle DCB$
hence their [[Definition:Area|areas]] are equal.
The [[Area of Parallelogram]] is equal t... | Area of Triangle in Terms of Side and Altitude/Proof 1 | https://proofwiki.org/wiki/Area_of_Triangle_in_Terms_of_Side_and_Altitude | https://proofwiki.org/wiki/Area_of_Triangle_in_Terms_of_Side_and_Altitude/Proof_1 | [
"Area of Triangle in Terms of Side and Altitude",
"Areas of Triangles"
] | [
"Definition:Area",
"Definition:Triangle (Geometry)",
"Definition:Polygon/Side",
"Definition:Altitude of Triangle"
] | [
"File:Area-of-Triangle.png",
"Definition:Quadrilateral/Parallelogram",
"Opposite Sides and Angles of Parallelogram are Equal",
"Definition:Area",
"Area of Parallelogram",
"Definition:Multiplication/Real Numbers",
"Definition:Parallelogram/Base",
"Definition:Parallelogram/Altitude",
"Axiom:Area Axiom... |
proofwiki-1509 | Absolutely Convergent Series is Convergent | Let $V$ be a Banach space with norm $\norm {\, \cdot \,}$.
Let $\ds \sum_{n \mathop = 1}^\infty a_n$ be an absolutely convergent series in $V$.
Then $\ds \sum_{n \mathop = 1}^\infty a_n$ is convergent. | That $\ds \sum_{n \mathop = 1}^\infty a_n$ is absolutely convergent means that $\ds \sum_{n \mathop = 1}^\infty \norm {a_n}$ converges in $\R$.
Hence the sequence of partial sums is a Cauchy sequence by Convergent Sequence is Cauchy Sequence.
Now let $\epsilon > 0$.
Let $N \in \N$ such that for all $m, n \in \N$, $m \g... | Let $V$ be a [[Definition:Banach Space|Banach space]] with [[Definition:Norm on Vector Space|norm]] $\norm {\, \cdot \,}$.
Let $\ds \sum_{n \mathop = 1}^\infty a_n$ be an [[Definition:Absolutely Convergent Series|absolutely convergent series]] in $V$.
Then $\ds \sum_{n \mathop = 1}^\infty a_n$ is [[Definition:Conver... | That $\ds \sum_{n \mathop = 1}^\infty a_n$ is [[Definition:Absolutely Convergent Series|absolutely convergent]] means that $\ds \sum_{n \mathop = 1}^\infty \norm {a_n}$ [[Definition:Convergent Series|converges]] in $\R$.
Hence the sequence of [[Definition:Partial Sum|partial sums]] is a [[Definition:Cauchy Sequence|Ca... | Absolutely Convergent Series is Convergent | https://proofwiki.org/wiki/Absolutely_Convergent_Series_is_Convergent | https://proofwiki.org/wiki/Absolutely_Convergent_Series_is_Convergent | [
"Absolutely Convergent Series is Convergent",
"Absolute Convergence",
"Banach Spaces",
"Convergence Tests"
] | [
"Definition:Banach Space",
"Definition:Norm/Vector Space",
"Definition:Absolutely Convergent Series",
"Definition:Convergent Series"
] | [
"Definition:Absolutely Convergent Series",
"Definition:Convergent Series",
"Definition:Series/Sequence of Partial Sums",
"Definition:Cauchy Sequence",
"Convergent Sequence is Cauchy Sequence",
"Definition:Cauchy Sequence",
"Definition:Norm/Vector Space",
"Definition:Series",
"Definition:Cauchy Seque... |
proofwiki-1510 | Limit Comparison Test | Let $\sequence {a_n}$ and $\sequence {b_n}$ be sequences in $\R$.
Let $\ds \frac {a_n} {b_n} \to l$ as $n \to \infty$ where $l \in \R_{>0}$.
Then the series $\ds \sum_{n \mathop = 1}^\infty a_n$ and $\ds \sum_{n \mathop = 1}^\infty b_n$ are either both convergent or both divergent. | Let $\ds \sum_{n \mathop = 1}^\infty b_n$ be convergent.
Then by Terms in Convergent Series Converge to Zero, $\sequence {b_n}$ converges to zero.
A Convergent Sequence is Bounded.
So it follows that:
:$\exists H: \forall n \in \N_{>0}: a_n \le H b_n$
Thus, by {{Corollary|Comparison Test|1}}, $\ds \sum_{n \mathop = 1}^... | Let $\sequence {a_n}$ and $\sequence {b_n}$ be [[Definition:Real Sequence|sequences in $\R$]].
Let $\ds \frac {a_n} {b_n} \to l$ as $n \to \infty$ where $l \in \R_{>0}$.
Then the [[Definition:Series|series]] $\ds \sum_{n \mathop = 1}^\infty a_n$ and $\ds \sum_{n \mathop = 1}^\infty b_n$ are either both [[Definition:... | Let $\ds \sum_{n \mathop = 1}^\infty b_n$ be [[Definition:Convergent Series|convergent]].
Then by [[Terms in Convergent Series Converge to Zero]], $\sequence {b_n}$ [[Definition:Convergent Sequence|converges]] to zero.
A [[Convergent Sequence is Bounded]].
So it follows that:
:$\exists H: \forall n \in \N_{>0}: a_n ... | Limit Comparison Test | https://proofwiki.org/wiki/Limit_Comparison_Test | https://proofwiki.org/wiki/Limit_Comparison_Test | [
"Convergence Tests",
"Series"
] | [
"Definition:Real Sequence",
"Definition:Series",
"Definition:Convergent Series",
"Definition:Divergent Series"
] | [
"Definition:Convergent Series",
"Terms in Convergent Series Converge to Zero",
"Definition:Convergent Sequence",
"Convergent Sequence in Metric Space is Bounded",
"Definition:Convergent Series",
"Sequence Converges to Within Half Limit"
] |
proofwiki-1511 | Area of Parallelogram | The area of a parallelogram equals the product of one of its bases and the associated altitude. | There are three cases to be analysed: the square, the rectangle and the general parallelogram. | The [[Definition:Area|area]] of a [[Definition:Parallelogram|parallelogram]] equals the product of one of its [[Definition:Base of Parallelogram|bases]] and the associated [[Definition:Altitude of Parallelogram|altitude]]. | There are three cases to be analysed: the [[Definition:Square (Geometry)|square]], the [[Definition:Rectangle|rectangle]] and the general [[Definition:Parallelogram|parallelogram]]. | Area of Parallelogram | https://proofwiki.org/wiki/Area_of_Parallelogram | https://proofwiki.org/wiki/Area_of_Parallelogram | [
"Area of Parallelogram",
"Parallelograms",
"Areas of Quadrilaterals",
"Area Formulas"
] | [
"Definition:Area",
"Definition:Quadrilateral/Parallelogram",
"Definition:Parallelogram/Base",
"Definition:Parallelogram/Altitude"
] | [
"Definition:Quadrilateral/Square",
"Definition:Quadrilateral/Rectangle",
"Definition:Quadrilateral/Parallelogram"
] |
proofwiki-1512 | Area of Triangle in Terms of Inradius and Exradii | Let $\triangle ABC$ be a triangle whose sides are $a$, $b$ and $c$ opposite vertices $A$, $B$ and $C$ respectively.
Then the area $\AA$ of $\triangle ABC$ is given by:
:$\AA = \sqrt {r \rho_a \rho_b \rho_c}$
where:
:$r$ is the inradius
:$\rho_a$, $\rho_b$ and $\rho_c$ are the exradii of $\triangle ABC$ {{WRT}} $a$, $b$... | {{begin-eqn}}
{{eqn | l = \AA
| r = \rho_a \paren {s - a}
| c = Area of Triangle in Terms of Exradius
}}
{{eqn | r = \rho_b \paren {s - b}
| c = Area of Triangle in Terms of Exradius
}}
{{eqn | r = \rho_c \paren {s - c}
| c = Area of Triangle in Terms of Exradius
}}
{{eqn | r = r s
| c = A... | Let $\triangle ABC$ be a [[Definition:Triangle (Geometry)|triangle]] whose [[Definition:Side of Polygon|sides]] are $a$, $b$ and $c$ [[Definition:Opposite (in Triangle)|opposite]] [[Definition:Vertex of Polygon|vertices]] $A$, $B$ and $C$ respectively.
Then the [[Definition:Area|area]] $\AA$ of $\triangle ABC$ is give... | {{begin-eqn}}
{{eqn | l = \AA
| r = \rho_a \paren {s - a}
| c = [[Area of Triangle in Terms of Exradius]]
}}
{{eqn | r = \rho_b \paren {s - b}
| c = [[Area of Triangle in Terms of Exradius]]
}}
{{eqn | r = \rho_c \paren {s - c}
| c = [[Area of Triangle in Terms of Exradius]]
}}
{{eqn | r = r s
... | Area of Triangle in Terms of Inradius and Exradii | https://proofwiki.org/wiki/Area_of_Triangle_in_Terms_of_Inradius_and_Exradii | https://proofwiki.org/wiki/Area_of_Triangle_in_Terms_of_Inradius_and_Exradii | [
"Excircles of Triangles",
"Areas of Triangles"
] | [
"Definition:Triangle (Geometry)",
"Definition:Polygon/Side",
"Definition:Triangle (Geometry)/Opposite",
"Definition:Polygon/Vertex",
"Definition:Area",
"Definition:Incircle of Triangle/Inradius",
"Definition:Excircle of Triangle/Exradius"
] | [
"Area of Triangle in Terms of Exradius",
"Area of Triangle in Terms of Exradius",
"Area of Triangle in Terms of Exradius",
"Area of Triangle in Terms of Inradius",
"Heron's Formula"
] |
proofwiki-1513 | Stewart's Theorem | Let $\triangle ABC$ be a triangle with sides $a, b, c$.
Let $CP$ be a cevian from $C$ to $P$.
:400px
Then:
:$a^2 \cdot AP + b^2 \cdot PB = c \paren {CP^2 + AP \cdot PB}$ | {{begin-eqn}}
{{eqn | n = 1
| l = b^2
| r = AP^2 + CP^2 - 2 AP \cdot CP \cdot \map \cos {\angle APC}
| c = Law of Cosines
}}
{{eqn | n = 2
| l = a^2
| r = PB^2 + CP^2 - 2 CP \cdot PB \cdot \map \cos {\angle BPC}
| c = Law of Cosines
}}
{{eqn | r = PB^2 + CP^2 + 2 CP \cdot PB \cdot \m... | Let $\triangle ABC$ be a [[Definition:Triangle (Geometry)|triangle]] with [[Definition:Side of Polygon|sides]] $a, b, c$.
Let $CP$ be a [[Definition:Cevian|cevian]] from $C$ to $P$.
:[[File:Stewart's Theorem.png|400px]]
Then:
:$a^2 \cdot AP + b^2 \cdot PB = c \paren {CP^2 + AP \cdot PB}$ | {{begin-eqn}}
{{eqn | n = 1
| l = b^2
| r = AP^2 + CP^2 - 2 AP \cdot CP \cdot \map \cos {\angle APC}
| c = [[Law of Cosines]]
}}
{{eqn | n = 2
| l = a^2
| r = PB^2 + CP^2 - 2 CP \cdot PB \cdot \map \cos {\angle BPC}
| c = [[Law of Cosines]]
}}
{{eqn | r = PB^2 + CP^2 + 2 CP \cdot PB ... | Stewart's Theorem | https://proofwiki.org/wiki/Stewart's_Theorem | https://proofwiki.org/wiki/Stewart's_Theorem | [
"Stewart's Theorem",
"Triangles"
] | [
"Definition:Triangle (Geometry)",
"Definition:Polygon/Side",
"Definition:Cevian",
"File:Stewart's Theorem.png"
] | [
"Law of Cosines",
"Law of Cosines",
"Cosine of Supplementary Angle"
] |
proofwiki-1514 | Median Formula | Let $\triangle ABC$ be a triangle.
Let $CD$ be the median of $\triangle ABC$ which bisects $AB$.
:400px
The length $m_c$ of $CD$ is given by:
:${m_c}^2 = \dfrac {a^2 + b^2} 2 - \dfrac {c^2} 4$ | {{begin-eqn}}
{{eqn | l = a^2 \cdot AD + b^2 \cdot DB
| r = CD^2 \cdot c + AD \cdot DB \cdot c
| c = Stewart's Theorem
}}
{{eqn | ll= \leadsto
| l = a^2 \frac c 2 + b^2 \frac c 2
| r = {m_c}^2 \cdot c + \paren {\frac c 2}^2 c
| c = substituting $AD = DB = \dfrac c 2$ and $CD = m_c$
}}... | Let $\triangle ABC$ be a [[Definition:Triangle (Geometry)|triangle]].
Let $CD$ be the [[Definition:Median of Triangle|median]] of $\triangle ABC$ which [[Definition:Bisection|bisects]] $AB$.
:[[File:MedianOfTriangle.png|400px]]
The [[Definition:Length (Linear Measure)|length]] $m_c$ of $CD$ is given by:
:${m_c}^2 =... | {{begin-eqn}}
{{eqn | l = a^2 \cdot AD + b^2 \cdot DB
| r = CD^2 \cdot c + AD \cdot DB \cdot c
| c = [[Stewart's Theorem]]
}}
{{eqn | ll= \leadsto
| l = a^2 \frac c 2 + b^2 \frac c 2
| r = {m_c}^2 \cdot c + \paren {\frac c 2}^2 c
| c = substituting $AD = DB = \dfrac c 2$ and $CD = m_c... | Median Formula/Proof 1 | https://proofwiki.org/wiki/Median_Formula | https://proofwiki.org/wiki/Median_Formula/Proof_1 | [
"Median Formula",
"Medians of Triangles",
"Named Theorems"
] | [
"Definition:Triangle (Geometry)",
"Definition:Median of Triangle",
"Definition:Bisection",
"File:MedianOfTriangle.png",
"Definition:Linear Measure/Length"
] | [
"Stewart's Theorem"
] |
proofwiki-1515 | Median Formula | Let $\triangle ABC$ be a triangle.
Let $CD$ be the median of $\triangle ABC$ which bisects $AB$.
:400px
The length $m_c$ of $CD$ is given by:
:${m_c}^2 = \dfrac {a^2 + b^2} 2 - \dfrac {c^2} 4$ | Let $\triangle ABC$ be embedded in the complex plane.
:300px
Let $\mathbf a = \overrightarrow {AC}$ and $\mathbf b = \overrightarrow {BC}$.
Then:
{{begin-eqn}}
{{eqn | l = \overrightarrow {AB}
| r = \mathbf a - \mathbf b
| c =
}}
{{eqn | l = \overrightarrow {AD}
| r = \dfrac {\overrightarrow {AB} } 2... | Let $\triangle ABC$ be a [[Definition:Triangle (Geometry)|triangle]].
Let $CD$ be the [[Definition:Median of Triangle|median]] of $\triangle ABC$ which [[Definition:Bisection|bisects]] $AB$.
:[[File:MedianOfTriangle.png|400px]]
The [[Definition:Length (Linear Measure)|length]] $m_c$ of $CD$ is given by:
:${m_c}^2 =... | Let $\triangle ABC$ be embedded in the [[Definition:Complex Plane|complex plane]].
:[[File:Length-of-Triangle-Median-Complex.png|300px]]
Let $\mathbf a = \overrightarrow {AC}$ and $\mathbf b = \overrightarrow {BC}$.
Then:
{{begin-eqn}}
{{eqn | l = \overrightarrow {AB}
| r = \mathbf a - \mathbf b
| c =... | Median Formula/Proof 2 | https://proofwiki.org/wiki/Median_Formula | https://proofwiki.org/wiki/Median_Formula/Proof_2 | [
"Median Formula",
"Medians of Triangles",
"Named Theorems"
] | [
"Definition:Triangle (Geometry)",
"Definition:Median of Triangle",
"Definition:Bisection",
"File:MedianOfTriangle.png",
"Definition:Linear Measure/Length"
] | [
"Definition:Complex Number/Complex Plane",
"File:Length-of-Triangle-Median-Complex.png",
"Dot Product of Vector with Itself",
"Square of Sum of Vectors",
"Square of Sum of Vectors",
"Dot Product of Vector with Itself"
] |
proofwiki-1516 | Median Formula | Let $\triangle ABC$ be a triangle.
Let $CD$ be the median of $\triangle ABC$ which bisects $AB$.
:400px
The length $m_c$ of $CD$ is given by:
:${m_c}^2 = \dfrac {a^2 + b^2} 2 - \dfrac {c^2} 4$ | {{begin-eqn}}
{{eqn | l = {m_c}^2
| r = b^2 + \paren {\frac c 2}^2 - 2 b \paren {\frac c 2} \cos A
| c = Law of Cosines
}}
{{eqn | r = b^2 + \frac {c^2} 4 - 2 b \paren {\frac c 2} \paren {\frac {b^2 + c^2 - a^2} {2 b c} }
| c = Law of Cosines
}}
{{eqn | r = b^2 + \frac {c^2} 4 - \frac {b^2 + c^2 ... | Let $\triangle ABC$ be a [[Definition:Triangle (Geometry)|triangle]].
Let $CD$ be the [[Definition:Median of Triangle|median]] of $\triangle ABC$ which [[Definition:Bisection|bisects]] $AB$.
:[[File:MedianOfTriangle.png|400px]]
The [[Definition:Length (Linear Measure)|length]] $m_c$ of $CD$ is given by:
:${m_c}^2 =... | {{begin-eqn}}
{{eqn | l = {m_c}^2
| r = b^2 + \paren {\frac c 2}^2 - 2 b \paren {\frac c 2} \cos A
| c = [[Law of Cosines]]
}}
{{eqn | r = b^2 + \frac {c^2} 4 - 2 b \paren {\frac c 2} \paren {\frac {b^2 + c^2 - a^2} {2 b c} }
| c = [[Law of Cosines]]
}}
{{eqn | r = b^2 + \frac {c^2} 4 - \frac {b^... | Median Formula/Proof 3 | https://proofwiki.org/wiki/Median_Formula | https://proofwiki.org/wiki/Median_Formula/Proof_3 | [
"Median Formula",
"Medians of Triangles",
"Named Theorems"
] | [
"Definition:Triangle (Geometry)",
"Definition:Median of Triangle",
"Definition:Bisection",
"File:MedianOfTriangle.png",
"Definition:Linear Measure/Length"
] | [
"Law of Cosines",
"Law of Cosines"
] |
proofwiki-1517 | Length of Angle Bisector | Let $\triangle ABC$ be a triangle.
Let $AD$ be the angle bisector of $\angle BAC$ in $\triangle ABC$.
:300px
Let $d$ be the length of $AD$.
Then $d$ is given by:
:$d^2 = \dfrac {b c} {\paren {b + c}^2} \paren {\paren {b + c}^2 - a^2}$
where $a$, $b$, and $c$ are the sides opposite $A$, $B$ and $C$ respectively. | {{begin-eqn}}
{{eqn | l = \frac {BD} {DC}
| r = \frac c b
| c = Angle Bisector Theorem
}}
{{eqn | ll= \leadsto
| l = \frac {BD} {DC} + 1
| r = \frac c b + 1
}}
{{eqn | ll= \leadsto
| l = \frac {BD + DC} {DC}
| r = \frac {b + c} b
}}
{{eqn | ll= \leadsto
| l = \frac a {DC}
... | Let $\triangle ABC$ be a [[Definition:Triangle (Geometry)|triangle]].
Let $AD$ be the [[Definition:Angle Bisector|angle bisector]] of $\angle BAC$ in $\triangle ABC$.
:[[File:LengthOfAngleBisector.png|300px]]
Let $d$ be the [[Definition:Length of Line|length]] of $AD$.
Then $d$ is given by:
:$d^2 = \dfrac {b c} {... | {{begin-eqn}}
{{eqn | l = \frac {BD} {DC}
| r = \frac c b
| c = [[Angle Bisector Theorem]]
}}
{{eqn | ll= \leadsto
| l = \frac {BD} {DC} + 1
| r = \frac c b + 1
}}
{{eqn | ll= \leadsto
| l = \frac {BD + DC} {DC}
| r = \frac {b + c} b
}}
{{eqn | ll= \leadsto
| l = \frac a {DC}
... | Length of Angle Bisector/Proof 1 | https://proofwiki.org/wiki/Length_of_Angle_Bisector | https://proofwiki.org/wiki/Length_of_Angle_Bisector/Proof_1 | [
"Length of Angle Bisector",
"Triangles",
"Angle Bisectors"
] | [
"Definition:Triangle (Geometry)",
"Definition:Angle Bisector",
"File:LengthOfAngleBisector.png",
"Definition:Linear Measure/Length",
"Definition:Triangle (Geometry)/Opposite"
] | [
"Angle Bisector Theorem",
"Stewart's Theorem"
] |
proofwiki-1518 | Length of Angle Bisector | Let $\triangle ABC$ be a triangle.
Let $AD$ be the angle bisector of $\angle BAC$ in $\triangle ABC$.
:300px
Let $d$ be the length of $AD$.
Then $d$ is given by:
:$d^2 = \dfrac {b c} {\paren {b + c}^2} \paren {\paren {b + c}^2 - a^2}$
where $a$, $b$, and $c$ are the sides opposite $A$, $B$ and $C$ respectively. | :400px
From Length of Angle Bisector: Proof 1, we have:
:$BD = \dfrac {a c} {b + c}$
:$DC = \dfrac {a b} {b + c}$
Then we have:
{{begin-eqn}}
{{eqn | l = \angle BAD
| o = \cong
| r = \angle FAC
| c = {{Defof|Angle Bisector}}
}}
{{eqn | l = \angle ABD
| o = \cong
| r = \angle AFC
| c ... | Let $\triangle ABC$ be a [[Definition:Triangle (Geometry)|triangle]].
Let $AD$ be the [[Definition:Angle Bisector|angle bisector]] of $\angle BAC$ in $\triangle ABC$.
:[[File:LengthOfAngleBisector.png|300px]]
Let $d$ be the [[Definition:Length of Line|length]] of $AD$.
Then $d$ is given by:
:$d^2 = \dfrac {b c} {... | :[[File:LengthOfAngleBisector2.png|400px]]
From [[Length of Angle Bisector/Proof 1|Length of Angle Bisector: Proof 1]], we have:
:$BD = \dfrac {a c} {b + c}$
:$DC = \dfrac {a b} {b + c}$
Then we have:
{{begin-eqn}}
{{eqn | l = \angle BAD
| o = \cong
| r = \angle FAC
| c = {{Defof|Angle Bisector}}... | Length of Angle Bisector/Proof 2 | https://proofwiki.org/wiki/Length_of_Angle_Bisector | https://proofwiki.org/wiki/Length_of_Angle_Bisector/Proof_2 | [
"Length of Angle Bisector",
"Triangles",
"Angle Bisectors"
] | [
"Definition:Triangle (Geometry)",
"Definition:Angle Bisector",
"File:LengthOfAngleBisector.png",
"Definition:Linear Measure/Length",
"Definition:Triangle (Geometry)/Opposite"
] | [
"File:LengthOfAngleBisector2.png",
"Length of Angle Bisector/Proof 1",
"Angles in Same Segment of Circle are Equal",
"Triangles with Two Equal Angles are Similar",
"Definition:Similar Triangles",
"Intersecting Chords Theorem"
] |
proofwiki-1519 | Supremum of Subset | Let $\left({U, \preceq}\right)$ be an ordered set.
Let $S \subseteq U$.
Let $T \subseteq S$.
Let $S$ admit a supremum (in $U$).
If $T$ also admits a supremum (in $U$), then $\sup \left({T}\right) \preceq\sup \left({S}\right)$. | The number $\sup S$ is an upper bound for $S$.
Therefore, $\sup S$ is an upper bound for $T$ as $T$ is a non-empty subset of $S$.
Accordingly, $T$ has a supremum by the Continuum Property.
The number $\sup S$ is an upper bound for $T$.
Therefore, $\sup S$ is greater than or equal to $\sup T$ as $\sup T$ is the least up... | Let $\left({U, \preceq}\right)$ be an [[Definition:Ordered Set|ordered set]].
Let $S \subseteq U$.
Let $T \subseteq S$.
Let $S$ admit a [[Definition:Supremum of Set|supremum]] (in $U$).
If $T$ also admits a [[Definition:Supremum of Set|supremum]] (in $U$), then $\sup \left({T}\right) \preceq\sup \left({S}\right)$. | The [[Definition:Real Number|number]] $\sup S$ is an [[Definition:Upper Bound of Subset of Real Numbers|upper bound]] for $S$.
Therefore, $\sup S$ is an [[Definition:Upper Bound of Subset of Real Numbers|upper bound]] for $T$ as $T$ is a [[Definition:Non-Empty Set|non-empty]] [[Definition:Subset|subset]] of $S$.
Acco... | Supremum of Set of Real Numbers is at least Supremum of Subset/Proof 1 | https://proofwiki.org/wiki/Supremum_of_Subset | https://proofwiki.org/wiki/Supremum_of_Set_of_Real_Numbers_is_at_least_Supremum_of_Subset/Proof_1 | [
"Order Theory"
] | [
"Definition:Ordered Set",
"Definition:Supremum of Set",
"Definition:Supremum of Set"
] | [
"Definition:Real Number",
"Definition:Upper Bound of Set/Real Numbers",
"Definition:Upper Bound of Set/Real Numbers",
"Definition:Non-Empty Set",
"Definition:Subset",
"Definition:Supremum of Set/Real Numbers",
"Continuum Property",
"Definition:Real Number",
"Definition:Upper Bound of Set/Real Number... |
proofwiki-1520 | Supremum of Subset | Let $\left({U, \preceq}\right)$ be an ordered set.
Let $S \subseteq U$.
Let $T \subseteq S$.
Let $S$ admit a supremum (in $U$).
If $T$ also admits a supremum (in $U$), then $\sup \left({T}\right) \preceq\sup \left({S}\right)$. | By the Continuum Property, $T$ admits a supremum.
It follows from Supremum of Subset that $\sup T \le \sup S$.
{{qed}} | Let $\left({U, \preceq}\right)$ be an [[Definition:Ordered Set|ordered set]].
Let $S \subseteq U$.
Let $T \subseteq S$.
Let $S$ admit a [[Definition:Supremum of Set|supremum]] (in $U$).
If $T$ also admits a [[Definition:Supremum of Set|supremum]] (in $U$), then $\sup \left({T}\right) \preceq\sup \left({S}\right)$. | By the [[Continuum Property]], $T$ admits a [[Definition:Supremum of Subset of Real Numbers|supremum]].
It follows from [[Supremum of Subset]] that $\sup T \le \sup S$.
{{qed}} | Supremum of Set of Real Numbers is at least Supremum of Subset/Proof 2 | https://proofwiki.org/wiki/Supremum_of_Subset | https://proofwiki.org/wiki/Supremum_of_Set_of_Real_Numbers_is_at_least_Supremum_of_Subset/Proof_2 | [
"Order Theory"
] | [
"Definition:Ordered Set",
"Definition:Supremum of Set",
"Definition:Supremum of Set"
] | [
"Continuum Property",
"Definition:Supremum of Set/Real Numbers",
"Supremum of Subset"
] |
proofwiki-1521 | Supremum of Subset | Let $\left({U, \preceq}\right)$ be an ordered set.
Let $S \subseteq U$.
Let $T \subseteq S$.
Let $S$ admit a supremum (in $U$).
If $T$ also admits a supremum (in $U$), then $\sup \left({T}\right) \preceq\sup \left({S}\right)$. | $S$ is bounded above as $S$ has a supremum.
Therefore, $T$ is bounded above as $T$ is a subset of $S$.
Accordingly, $T$ admits a supremum by the Continuum Property as $T$ is non-empty.
We know that $\sup T$ and $\sup S$ exist.
Therefore by Suprema of two Real Sets:
:$\forall \epsilon \in \R_{>0}: \forall t \in T: \exis... | Let $\left({U, \preceq}\right)$ be an [[Definition:Ordered Set|ordered set]].
Let $S \subseteq U$.
Let $T \subseteq S$.
Let $S$ admit a [[Definition:Supremum of Set|supremum]] (in $U$).
If $T$ also admits a [[Definition:Supremum of Set|supremum]] (in $U$), then $\sup \left({T}\right) \preceq\sup \left({S}\right)$. | $S$ is [[Definition:Bounded Above Set|bounded above]] as $S$ has a [[Definition:Supremum of Subset of Real Numbers|supremum]].
Therefore, $T$ is [[Definition:Bounded Above Set|bounded above]] as $T$ is a [[Definition:Subset|subset]] of $S$.
Accordingly, $T$ admits a [[Definition:Supremum of Subset of Real Numbers|sup... | Supremum of Set of Real Numbers is at least Supremum of Subset/Proof 3 | https://proofwiki.org/wiki/Supremum_of_Subset | https://proofwiki.org/wiki/Supremum_of_Set_of_Real_Numbers_is_at_least_Supremum_of_Subset/Proof_3 | [
"Order Theory"
] | [
"Definition:Ordered Set",
"Definition:Supremum of Set",
"Definition:Supremum of Set"
] | [
"Definition:Bounded Above Set",
"Definition:Supremum of Set/Real Numbers",
"Definition:Bounded Above Set",
"Definition:Subset",
"Definition:Supremum of Set/Real Numbers",
"Continuum Property",
"Definition:Non-Empty Set",
"Suprema of two Real Sets"
] |
proofwiki-1522 | Supremum of Subset | Let $\left({U, \preceq}\right)$ be an ordered set.
Let $S \subseteq U$.
Let $T \subseteq S$.
Let $S$ admit a supremum (in $U$).
If $T$ also admits a supremum (in $U$), then $\sup \left({T}\right) \preceq\sup \left({S}\right)$. | By definition $\sup S$ is an upper bound for $S$.
Thus:
:$\forall x \in S: x \le \sup S$
As $T \subseteq S$ we have by definition of subset that:
:$\forall x \in T: x \in S$
Hence:
:$\forall x \in T: x \le \sup S$
So by definition $\sup S$ is an upper bound for $T$.
So $\sup S$ is at least as big as the smallest upper ... | Let $\left({U, \preceq}\right)$ be an [[Definition:Ordered Set|ordered set]].
Let $S \subseteq U$.
Let $T \subseteq S$.
Let $S$ admit a [[Definition:Supremum of Set|supremum]] (in $U$).
If $T$ also admits a [[Definition:Supremum of Set|supremum]] (in $U$), then $\sup \left({T}\right) \preceq\sup \left({S}\right)$. | By definition $\sup S$ is an [[Definition:Upper Bound|upper bound]] for $S$.
Thus:
:$\forall x \in S: x \le \sup S$
As $T \subseteq S$ we have by definition of [[Definition:Subset|subset]] that:
:$\forall x \in T: x \in S$
Hence:
:$\forall x \in T: x \le \sup S$
So by definition $\sup S$ is an [[Definition:Upper Bo... | Supremum of Set of Real Numbers is at least Supremum of Subset/Proof 4 | https://proofwiki.org/wiki/Supremum_of_Subset | https://proofwiki.org/wiki/Supremum_of_Set_of_Real_Numbers_is_at_least_Supremum_of_Subset/Proof_4 | [
"Order Theory"
] | [
"Definition:Ordered Set",
"Definition:Supremum of Set",
"Definition:Supremum of Set"
] | [
"Definition:Upper Bound",
"Definition:Subset",
"Definition:Upper Bound",
"Definition:Smallest Element",
"Definition:Upper Bound",
"Definition:Supremum"
] |
proofwiki-1523 | Supremum of Subset | Let $\left({U, \preceq}\right)$ be an ordered set.
Let $S \subseteq U$.
Let $T \subseteq S$.
Let $S$ admit a supremum (in $U$).
If $T$ also admits a supremum (in $U$), then $\sup \left({T}\right) \preceq\sup \left({S}\right)$. | Let $B = \sup \left({S}\right)$.
Then $B$ is an upper bound for $S$.
As $T \subseteq S$, it follows by the definition of a subset that $x \in T \implies x \in S$.
Because $x \in S \implies x \preceq B$ (as $B$ is an upper bound for $S$) it follows that $x \in T \implies x \preceq B$.
So $B$ is an upper bound for $T$.
T... | Let $\left({U, \preceq}\right)$ be an [[Definition:Ordered Set|ordered set]].
Let $S \subseteq U$.
Let $T \subseteq S$.
Let $S$ admit a [[Definition:Supremum of Set|supremum]] (in $U$).
If $T$ also admits a [[Definition:Supremum of Set|supremum]] (in $U$), then $\sup \left({T}\right) \preceq\sup \left({S}\right)$. | Let $B = \sup \left({S}\right)$.
Then $B$ is an [[Definition:Upper Bound of Set|upper bound]] for $S$.
As $T \subseteq S$, it follows by the definition of a [[Definition:Subset|subset]] that $x \in T \implies x \in S$.
Because $x \in S \implies x \preceq B$ (as $B$ is an [[Definition:Upper Bound of Set|upper bound]]... | Supremum of Subset | https://proofwiki.org/wiki/Supremum_of_Subset | https://proofwiki.org/wiki/Supremum_of_Subset | [
"Order Theory"
] | [
"Definition:Ordered Set",
"Definition:Supremum of Set",
"Definition:Supremum of Set"
] | [
"Definition:Upper Bound of Set",
"Definition:Subset",
"Definition:Upper Bound of Set",
"Definition:Upper Bound of Set",
"Definition:Ordering",
"Definition:Supremum of Set",
"Category:Order Theory"
] |
proofwiki-1524 | Area of Triangle | This page gathers a variety of formulas for the area of a triangle. | The point $\tuple {x = 0, y = 0}$ satisfies $y = x^2$.
Since $x^2 \ge 0$, $0$ is the minimum value for $y$.
Thus, the vertex of $T$ lies at the origin.
The coordinates of $C$ are given as the average those for $A$ and $B$.
Given:
:$A = \tuple {u, u^2}$
:$B = \tuple {v, v^2}$, with $u > v$
By the definition of average: ... | This page gathers a variety of formulas for the [[Definition:Area|area]] of a [[Definition:Triangle (Geometry)|triangle]]. | The [[Definition:Point|point]] $\tuple {x = 0, y = 0}$ satisfies $y = x^2$.
Since $x^2 \ge 0$, $0$ is the [[Definition:Minimum Value of Real Function|minimum value]] for $y$.
Thus, the [[Definition:Parabola|vertex]] of $T$ lies at the [[Definition:Origin of Coordinates|origin]].
The [[Definition:Homogeneous Cartesia... | Area of Triangle Inscribed in Parabola/Proof 1 | https://proofwiki.org/wiki/Area_of_Triangle | https://proofwiki.org/wiki/Area_of_Triangle_Inscribed_in_Parabola/Proof_1 | [
"Areas of Triangles"
] | [
"Definition:Area",
"Definition:Triangle (Geometry)"
] | [
"Definition:Point",
"Definition:Minimum Value of Real Function",
"Definition:Parabola",
"Definition:Coordinate System/Origin",
"Definition:Homogeneous Cartesian Coordinates",
"Definition:Arithmetic Mean",
"Definition:Arithmetic Mean",
"Definition:Line/Midpoint",
"Definition:Homogeneous Cartesian Coo... |
proofwiki-1525 | Area of Triangle | This page gathers a variety of formulas for the area of a triangle. | From Two-Point Form of Equation of Straight Line in Plane, the straight line $AB$ can be expressed as:
{{begin-eqn}}
{{eqn | l = \dfrac {\paren {y_{AB} - y_1} } {\paren {x - x_1} }
| r = \dfrac {\paren {y_2 - y_1} } {\paren {x_2 - x_1} }
| c =
}}
{{eqn | ll= \leadsto
| l = \dfrac {\paren {y_{AB} - v... | This page gathers a variety of formulas for the [[Definition:Area|area]] of a [[Definition:Triangle (Geometry)|triangle]]. | From [[Two-Point Form of Equation of Straight Line in Plane]], the [[Definition:Straight Line|straight line]] $AB$ can be expressed as:
{{begin-eqn}}
{{eqn | l = \dfrac {\paren {y_{AB} - y_1} } {\paren {x - x_1} }
| r = \dfrac {\paren {y_2 - y_1} } {\paren {x_2 - x_1} }
| c =
}}
{{eqn | ll= \leadsto
... | Area of Triangle Inscribed in Parabola/Proof 2 | https://proofwiki.org/wiki/Area_of_Triangle | https://proofwiki.org/wiki/Area_of_Triangle_Inscribed_in_Parabola/Proof_2 | [
"Areas of Triangles"
] | [
"Definition:Area",
"Definition:Triangle (Geometry)"
] | [
"Equation of Straight Line in Plane/Two-Point Form",
"Definition:Line/Straight Line",
"Difference of Two Squares",
"Definition:Division/Field/Real Numbers",
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Definition:Homogeneous Cartesian Coordinates",
"Definition:Line/Midpoint",
... |
proofwiki-1526 | Area of Triangle | This page gathers a variety of formulas for the area of a triangle. | {{begin-eqn}}
{{eqn | l = \map \Area T
| r = \dfrac 1 2 \size {\paren {\begin {vmatrix} 0 & 0 & 1 \\ b & a & 1 \\ x & y & 1 \end {vmatrix} } }
| c = Area of Triangle in Determinant Form
}}
{{eqn | r = \dfrac 1 2 \size {b y - a x}
| c = Determinant of Order 3
}}
{{end-eqn}}
{{qed}} | This page gathers a variety of formulas for the [[Definition:Area|area]] of a [[Definition:Triangle (Geometry)|triangle]]. | {{begin-eqn}}
{{eqn | l = \map \Area T
| r = \dfrac 1 2 \size {\paren {\begin {vmatrix} 0 & 0 & 1 \\ b & a & 1 \\ x & y & 1 \end {vmatrix} } }
| c = [[Area of Triangle in Determinant Form]]
}}
{{eqn | r = \dfrac 1 2 \size {b y - a x}
| c = [[Determinant of Order 3]]
}}
{{end-eqn}}
{{qed}} | Area of Triangle in Determinant Form with Vertex at Origin/Proof 1 | https://proofwiki.org/wiki/Area_of_Triangle | https://proofwiki.org/wiki/Area_of_Triangle_in_Determinant_Form_with_Vertex_at_Origin/Proof_1 | [
"Areas of Triangles"
] | [
"Definition:Area",
"Definition:Triangle (Geometry)"
] | [
"Area of Triangle in Determinant Form",
"Determinant/Examples/Order 3"
] |
proofwiki-1527 | Area of Triangle | This page gathers a variety of formulas for the area of a triangle. | Let the polar coordinates of $B$ and $C$ be:
{{begin-eqn}}
{{eqn | l = B
| r = \polar {r_1, \theta_1}
}}
{{eqn | l = C
| r = \polar {r_2, \theta_2}
}}
{{end-eqn}}
Let $\theta$ be the angle between $AB$ and $AC$.
Then we have:
{{begin-eqn}}
{{eqn | l = \map \Area {\triangle ABC}
| r = \dfrac 1 2 AB \cd... | This page gathers a variety of formulas for the [[Definition:Area|area]] of a [[Definition:Triangle (Geometry)|triangle]]. | Let the [[Definition:Polar Coordinates|polar coordinates]] of $B$ and $C$ be:
{{begin-eqn}}
{{eqn | l = B
| r = \polar {r_1, \theta_1}
}}
{{eqn | l = C
| r = \polar {r_2, \theta_2}
}}
{{end-eqn}}
Let $\theta$ be the [[Definition:Angle|angle]] between $AB$ and $AC$.
Then we have:
{{begin-eqn}}
{{eqn | l ... | Area of Triangle in Determinant Form with Vertex at Origin/Proof 2 | https://proofwiki.org/wiki/Area_of_Triangle | https://proofwiki.org/wiki/Area_of_Triangle_in_Determinant_Form_with_Vertex_at_Origin/Proof_2 | [
"Areas of Triangles"
] | [
"Definition:Area",
"Definition:Triangle (Geometry)"
] | [
"Definition:Polar Coordinates",
"Definition:Angle",
"Area of Triangle in Terms of Two Sides and Angle",
"Sine of Difference",
"Definition:Area",
"Definition:Positive/Real Number",
"Definition:Negative/Real Number",
"Definition:Sign of Number",
"Definition:Absolute Value"
] |
proofwiki-1528 | Area of Triangle | This page gathers a variety of formulas for the area of a triangle. | :400px
Let $A$, $B$ and $C$ be defined as complex numbers in the complex plane.
The vectors from $C$ to $A$ and from $C$ to $B$ are given by:
:$z_1 = \paren {x_1 - x_3} + i \paren {y_1 - y_3}$
:$z_2 = \paren {x_2 - x_3} + i \paren {y_2 - y_3}$
From Area of Triangle in Terms of Side and Altitude, $\AA$ is half that of a... | This page gathers a variety of formulas for the [[Definition:Area|area]] of a [[Definition:Triangle (Geometry)|triangle]]. | :[[File:AreaOfTriangleComplex.png|400px]]
Let $A$, $B$ and $C$ be defined as [[Definition:Complex Number|complex numbers]] in the [[Definition:Complex Plane|complex plane]].
The [[Definition:Complex Number as Vector|vectors]] from $C$ to $A$ and from $C$ to $B$ are given by:
:$z_1 = \paren {x_1 - x_3} + i \paren {y_... | Area of Triangle in Determinant Form/Proof 1 | https://proofwiki.org/wiki/Area_of_Triangle | https://proofwiki.org/wiki/Area_of_Triangle_in_Determinant_Form/Proof_1 | [
"Areas of Triangles"
] | [
"Definition:Area",
"Definition:Triangle (Geometry)"
] | [
"File:AreaOfTriangleComplex.png",
"Definition:Complex Number",
"Definition:Complex Number/Complex Plane",
"Definition:Complex Number as Vector",
"Area of Triangle in Terms of Side and Altitude",
"Definition:Quadrilateral/Parallelogram",
"Area of Parallelogram in Complex Plane",
"Determinant/Examples/O... |
proofwiki-1529 | Area of Triangle | This page gathers a variety of formulas for the area of a triangle. | :400px
Let $A$, $B$ and $C$ be as defined..
Let $O$ be the origin of the Cartesian plane in which $\triangle ABC$ is embedded.
Taking into account the signs of the areas of the various triangles involved:
:$\triangle ABC = \triangle OAB + \triangle OBC + \triangle OCA$
as it is seen that $\triangle OBC$ and $\triangle ... | This page gathers a variety of formulas for the [[Definition:Area|area]] of a [[Definition:Triangle (Geometry)|triangle]]. | :[[File:Area-of-Triangle-Determinant.png|400px]]
Let $A$, $B$ and $C$ be as defined..
Let $O$ be the [[Definition:Origin|origin]] of the [[Definition:Cartesian Plane|Cartesian plane]] in which $\triangle ABC$ is embedded.
Taking into account the [[Definition:Sign of Area of Triangle|signs]] of the [[Definition:Area... | Area of Triangle in Determinant Form/Proof 2 | https://proofwiki.org/wiki/Area_of_Triangle | https://proofwiki.org/wiki/Area_of_Triangle_in_Determinant_Form/Proof_2 | [
"Areas of Triangles"
] | [
"Definition:Area",
"Definition:Triangle (Geometry)"
] | [
"File:Area-of-Triangle-Determinant.png",
"Definition:Coordinate System/Origin",
"Definition:Cartesian Plane",
"Definition:Sign of Area of Triangle",
"Definition:Area",
"Definition:Triangle (Geometry)",
"Definition:Clockwise",
"Area of Triangle in Determinant Form with Vertex at Origin/Proof 2",
"Det... |
proofwiki-1530 | Area of Triangle | This page gathers a variety of formulas for the area of a triangle. | :400px
Let $A$, $B$ and $C$ be defined as $\tuple {x_1, y_1}$, $\tuple {x_2, y_2}$ and $\tuple {x_3, y_3}$ respectively.
From the figure, we see that:
{{begin-eqn}}
{{eqn | l = \map \Area {ABC}
| r = \map \Area {PACR} + \map \Area {RCBQ} - \map \Area {PABQ}
| c =
}}
{{eqn | r = \dfrac {\paren {x_3 - x_1} \... | This page gathers a variety of formulas for the [[Definition:Area|area]] of a [[Definition:Triangle (Geometry)|triangle]]. | :[[File:Area-of-Triangle-Determinant-Proof-3.png|400px]]
Let $A$, $B$ and $C$ be defined as $\tuple {x_1, y_1}$, $\tuple {x_2, y_2}$ and $\tuple {x_3, y_3}$ respectively.
From the figure, we see that:
{{begin-eqn}}
{{eqn | l = \map \Area {ABC}
| r = \map \Area {PACR} + \map \Area {RCBQ} - \map \Area {PABQ}
... | Area of Triangle in Determinant Form/Proof 3 | https://proofwiki.org/wiki/Area_of_Triangle | https://proofwiki.org/wiki/Area_of_Triangle_in_Determinant_Form/Proof_3 | [
"Areas of Triangles"
] | [
"Definition:Area",
"Definition:Triangle (Geometry)"
] | [
"File:Area-of-Triangle-Determinant-Proof-3.png",
"Area of Trapezium",
"Determinant/Examples/Order 3"
] |
proofwiki-1531 | Area of Triangle | This page gathers a variety of formulas for the area of a triangle. | :500px
Let $C$ be the excircle of $\triangle ABC$ which is tangent to $a$.
By definition:
:$\rho_a$ is the radius of $C$
:$I_a$ is the center of $C$.
Then we have:
{{begin-eqn}}
{{eqn | l = \AA
| r = \map \Area {\triangle ABI_a} + \map \Area {\triangle ACI_a} - \map \Area {\triangle CBI_a}
| c = (see figure... | This page gathers a variety of formulas for the [[Definition:Area|area]] of a [[Definition:Triangle (Geometry)|triangle]]. | :[[File:Area-of-Triangle-by-Exradius.png|500px]]
Let $C$ be the [[Definition:Excircle of Triangle|excircle]] of $\triangle ABC$ which is [[Definition:Tangent to Circle|tangent]] to $a$.
By definition:
:$\rho_a$ is the [[Definition:Radius of Circle|radius]] of $C$
:$I_a$ is the [[Definition:Center of Circle|center]] ... | Area of Triangle in Terms of Exradius/Proof | https://proofwiki.org/wiki/Area_of_Triangle | https://proofwiki.org/wiki/Area_of_Triangle_in_Terms_of_Exradius/Proof | [
"Areas of Triangles"
] | [
"Definition:Area",
"Definition:Triangle (Geometry)"
] | [
"File:Area-of-Triangle-by-Exradius.png",
"Definition:Excircle of Triangle",
"Definition:Tangent Line/Circle",
"Definition:Circle/Radius",
"Definition:Circle/Center",
"Area of Triangle in Terms of Side and Altitude"
] |
proofwiki-1532 | Area of Triangle | This page gathers a variety of formulas for the area of a triangle. | :400px
Construct a point $D$ so that $\Box ABDC$ is a parallelogram.
From Opposite Sides and Angles of Parallelogram are Equal:
:$\triangle ABC \cong \triangle DCB$
hence their areas are equal.
The Area of Parallelogram is equal to the product of one of its bases and the associated altitude.
Thus
{{begin-eqn}}
{{eqn |... | This page gathers a variety of formulas for the [[Definition:Area|area]] of a [[Definition:Triangle (Geometry)|triangle]]. | :[[File:Area-of-Triangle.png|400px]]
Construct a point $D$ so that $\Box ABDC$ is a [[Definition:Parallelogram|parallelogram]].
From [[Opposite Sides and Angles of Parallelogram are Equal]]:
:$\triangle ABC \cong \triangle DCB$
hence their [[Definition:Area|areas]] are equal.
The [[Area of Parallelogram]] is equal t... | Area of Triangle in Terms of Side and Altitude/Proof 1 | https://proofwiki.org/wiki/Area_of_Triangle | https://proofwiki.org/wiki/Area_of_Triangle_in_Terms_of_Side_and_Altitude/Proof_1 | [
"Areas of Triangles"
] | [
"Definition:Area",
"Definition:Triangle (Geometry)"
] | [
"File:Area-of-Triangle.png",
"Definition:Quadrilateral/Parallelogram",
"Opposite Sides and Angles of Parallelogram are Equal",
"Definition:Area",
"Area of Parallelogram",
"Definition:Multiplication/Real Numbers",
"Definition:Parallelogram/Base",
"Definition:Parallelogram/Altitude",
"Axiom:Area Axiom... |
proofwiki-1533 | Area of Triangle | This page gathers a variety of formulas for the area of a triangle. | :420px
{{begin-eqn}}
{{eqn | l = \map \Area {ABC}
| r = \frac 1 2 h c
| c = Area of Triangle in Terms of Side and Altitude
}}
{{eqn | r = \frac 1 2 h \paren {p + q}
}}
{{eqn | r = \frac 1 2 a b \paren {\frac p a \frac h b + \frac h a \frac q b}
}}
{{eqn | r = \frac 1 2 a b \paren {\sin \alpha \cos \beta + ... | This page gathers a variety of formulas for the [[Definition:Area|area]] of a [[Definition:Triangle (Geometry)|triangle]]. | :[[File:TriangleAreaTwoSidesAngle.png|420px]]
{{begin-eqn}}
{{eqn | l = \map \Area {ABC}
| r = \frac 1 2 h c
| c = [[Area of Triangle in Terms of Side and Altitude]]
}}
{{eqn | r = \frac 1 2 h \paren {p + q}
}}
{{eqn | r = \frac 1 2 a b \paren {\frac p a \frac h b + \frac h a \frac q b}
}}
{{eqn | r = \fr... | Area of Triangle in Terms of Two Sides and Angle/Proof 1 | https://proofwiki.org/wiki/Area_of_Triangle | https://proofwiki.org/wiki/Area_of_Triangle_in_Terms_of_Two_Sides_and_Angle/Proof_1 | [
"Areas of Triangles"
] | [
"Definition:Area",
"Definition:Triangle (Geometry)"
] | [
"File:TriangleAreaTwoSidesAngle.png",
"Area of Triangle in Terms of Side and Altitude",
"Sine of Sum"
] |
proofwiki-1534 | Area of Triangle | This page gathers a variety of formulas for the area of a triangle. | :400px
By definition of sine:
:$h = b \sin C$
From Area of Triangle in Terms of Side and Altitude:
:$\map \Area {ABC} = \dfrac {a h} 2$
Substituting:
:$\map \Area {ABC} = \dfrac {a b \sin C} 2$
{{qed}} | This page gathers a variety of formulas for the [[Definition:Area|area]] of a [[Definition:Triangle (Geometry)|triangle]]. | :[[File:TriangleAreaTwoSidesAngle-2.png|400px]]
By definition of [[Definition:Sine|sine]]:
:$h = b \sin C$
From [[Area of Triangle in Terms of Side and Altitude]]:
:$\map \Area {ABC} = \dfrac {a h} 2$
Substituting:
:$\map \Area {ABC} = \dfrac {a b \sin C} 2$
{{qed}} | Area of Triangle in Terms of Two Sides and Angle/Proof 2 | https://proofwiki.org/wiki/Area_of_Triangle | https://proofwiki.org/wiki/Area_of_Triangle_in_Terms_of_Two_Sides_and_Angle/Proof_2 | [
"Areas of Triangles"
] | [
"Definition:Area",
"Definition:Triangle (Geometry)"
] | [
"File:TriangleAreaTwoSidesAngle-2.png",
"Definition:Sine",
"Area of Triangle in Terms of Side and Altitude"
] |
proofwiki-1535 | Intersecting Chords Theorem | Let $AC$ and $BD$ both be chords of the same circle.
Let $AC$ and $BD$ intersect at $E$.
Then $AE \cdot EC = DE \cdot EB$. | :320px
Let $AC$ and $BD$ be intersecting chords of circle $ABCD$.
Let the point of intersection be $E$.
If $E$ is the center of $ABCD$ the solution is trivial, as $AE = EC = BE = ED$ and so $AE \cdot EC = BE \cdot ED$.
Otherwise, let $F$ be the center of $ABCD$.
Let $FG$ be drawn perpendicular to $AC$, and $FH$ be draw... | Let $AC$ and $BD$ both be [[Definition:Chord of Circle|chords]] of the same [[Definition:Circle|circle]].
Let $AC$ and $BD$ [[Definition:Intersection (Geometry)|intersect]] at $E$.
Then $AE \cdot EC = DE \cdot EB$. | :[[File:Euclid-III-35.png|320px]]
Let $AC$ and $BD$ be intersecting [[Definition:Chord of Circle|chords]] of [[Definition:Circle|circle]] $ABCD$.
Let the point of intersection be $E$.
If $E$ is the [[Definition:Center of Circle|center]] of $ABCD$ the solution is trivial, as $AE = EC = BE = ED$ and so $AE \cdot EC = ... | Intersecting Chords Theorem/Proof 1 | https://proofwiki.org/wiki/Intersecting_Chords_Theorem | https://proofwiki.org/wiki/Intersecting_Chords_Theorem/Proof_1 | [
"Intersecting Chords Theorem",
"Circles",
"Named Theorems"
] | [
"Definition:Circle/Chord",
"Definition:Circle",
"Definition:Intersection (Geometry)"
] | [
"File:Euclid-III-35.png",
"Definition:Circle/Chord",
"Definition:Circle",
"Definition:Circle/Center",
"Definition:Circle/Center",
"Perpendicular through Given Point",
"Perpendicular through Given Point",
"Conditions for Diameter to be Perpendicular Bisector",
"Definition:Bisection",
"Definition:Bi... |
proofwiki-1536 | Intersecting Chords Theorem | Let $AC$ and $BD$ both be chords of the same circle.
Let $AC$ and $BD$ intersect at $E$.
Then $AE \cdot EC = DE \cdot EB$. | Join $A$ with $B$ and $C$ with $D$, as shown in this diagram:
:310px
Then we have:
{{begin-eqn}}
{{eqn | l = \angle AEB
| o = \cong
| r = \angle DEC
| c = Two Straight Lines make Equal Opposite Angles
}}
{{eqn | l = \angle BAE
| o = \cong
| r = \angle CDE
| c = Angles in Same Segment... | Let $AC$ and $BD$ both be [[Definition:Chord of Circle|chords]] of the same [[Definition:Circle|circle]].
Let $AC$ and $BD$ [[Definition:Intersection (Geometry)|intersect]] at $E$.
Then $AE \cdot EC = DE \cdot EB$. | Join $A$ with $B$ and $C$ with $D$, as shown in this diagram:
:[[File:Euclid-III-35-2.png|310px]]
Then we have:
{{begin-eqn}}
{{eqn | l = \angle AEB
| o = \cong
| r = \angle DEC
| c = [[Two Straight Lines make Equal Opposite Angles]]
}}
{{eqn | l = \angle BAE
| o = \cong
| r = \angle CD... | Intersecting Chords Theorem/Proof 2 | https://proofwiki.org/wiki/Intersecting_Chords_Theorem | https://proofwiki.org/wiki/Intersecting_Chords_Theorem/Proof_2 | [
"Intersecting Chords Theorem",
"Circles",
"Named Theorems"
] | [
"Definition:Circle/Chord",
"Definition:Circle",
"Definition:Intersection (Geometry)"
] | [
"File:Euclid-III-35-2.png",
"Two Straight Lines make Equal Opposite Angles",
"Angles in Same Segment of Circle are Equal",
"Triangles with Two Equal Angles are Similar"
] |
proofwiki-1537 | Supremum Plus Constant | Let $S$ be a subset of the set of real numbers $\R$.
Let $S$ be bounded above.
Let $\xi \in \R$.
Then:
:$\ds \map {\sup_{x \mathop \in S} } {x + \xi} = \xi + \map {\sup_{x \mathop \in S} } x$
where $\sup$ denotes supremum. | Let $B = \sup S$.
Let $T = \set {x + \xi: x \in S}$.
Since $\forall x \in S: x \le B$ it follows that:
:$\forall x \in S: x + \xi \le B + \xi$
Hence $\xi + B$ is an upper bound for $T$.
If $C$ is the supremum for $T$ then $C \le \xi + B$.
On the other hand:
:$\forall y \in T: y \le C$
Therefore:
:$\forall y \in T: y - ... | Let $S$ be a [[Definition:Subset|subset]] of the [[Definition:Real Number|set of real numbers $\R$]].
Let $S$ be [[Definition:Bounded Above Set|bounded above]].
Let $\xi \in \R$.
Then:
:$\ds \map {\sup_{x \mathop \in S} } {x + \xi} = \xi + \map {\sup_{x \mathop \in S} } x$
where $\sup$ denotes [[Definition:Supremum... | Let $B = \sup S$.
Let $T = \set {x + \xi: x \in S}$.
Since $\forall x \in S: x \le B$ it follows that:
:$\forall x \in S: x + \xi \le B + \xi$
Hence $\xi + B$ is an [[Definition:Upper Bound of Set|upper bound]] for $T$.
If $C$ is the [[Definition:Supremum of Set|supremum]] for $T$ then $C \le \xi + B$.
On the oth... | Supremum Plus Constant | https://proofwiki.org/wiki/Supremum_Plus_Constant | https://proofwiki.org/wiki/Supremum_Plus_Constant | [
"Real Analysis"
] | [
"Definition:Subset",
"Definition:Real Number",
"Definition:Bounded Above Set",
"Definition:Supremum of Set"
] | [
"Definition:Upper Bound of Set",
"Definition:Supremum of Set",
"Definition:Upper Bound of Set"
] |
proofwiki-1538 | Multiple of Infimum | Let $T \subseteq \R: T \ne \O$ be a non-empty subset of the set of real numbers $\R$.
Let $T$ be bounded below.
Let $z \in \R: z > 0$ be a (strictly) positive real number.
Then:
:$\ds \map {\inf_{x \mathop \in T} } {z x} = z \map {\inf_{x \mathop \in T} } x$
where $\inf$ denotes infimum. | From Negative of Infimum is Supremum of Negatives:
:$\ds -\inf_{x \mathop \in T} x = \map {\sup_{x \mathop \in T} } {-x} \implies \inf_{x \mathop \in T} x = -\map {\sup_{x \mathop \in T} } {-x}$
Let $S = \set {x \in \R: -x \in T}$.
From Negative of Infimum is Supremum of Negatives, $S$ is bounded above.
From Multiple o... | Let $T \subseteq \R: T \ne \O$ be a [[Definition:Non-Empty Set|non-empty]] [[Definition:Subset|subset]] of the [[Definition:Real Number|set of real numbers $\R$]].
Let $T$ be [[Definition:Bounded Below Set|bounded below]].
Let $z \in \R: z > 0$ be a [[Definition:Strictly Positive Real Number|(strictly) positive real ... | From [[Negative of Infimum is Supremum of Negatives]]:
:$\ds -\inf_{x \mathop \in T} x = \map {\sup_{x \mathop \in T} } {-x} \implies \inf_{x \mathop \in T} x = -\map {\sup_{x \mathop \in T} } {-x}$
Let $S = \set {x \in \R: -x \in T}$.
From [[Negative of Infimum is Supremum of Negatives]], $S$ is [[Definition:Bounded... | Multiple of Infimum | https://proofwiki.org/wiki/Multiple_of_Infimum | https://proofwiki.org/wiki/Multiple_of_Infimum | [
"Real Analysis"
] | [
"Definition:Non-Empty Set",
"Definition:Subset",
"Definition:Real Number",
"Definition:Bounded Below Set",
"Definition:Strictly Positive/Real Number",
"Definition:Infimum of Set"
] | [
"Negative of Infimum is Supremum of Negatives",
"Negative of Infimum is Supremum of Negatives",
"Definition:Bounded Above Set",
"Multiple of Supremum"
] |
proofwiki-1539 | Law of Sines | Let $\triangle ABC$ be a triangle whose sides $a, b, c$ are such that $a$ is opposite $A$, $b$ is opposite $B$ and $c$ is opposite $C$.
Then:
:$\dfrac a {\sin A} = \dfrac b {\sin B} = \dfrac c {\sin C} = 2 R$
where $R$ is the circumradius of $\triangle ABC$. | Construct the altitude from $B$.
:240px
From the definition of sine:
:$\sin A = \dfrac h c$ and $\sin C = \dfrac h a$
Thus:
:$h = c \sin A$
and:
:$h = a \sin C$
This gives:
:$c \sin A = a \sin C$
So:
:$\dfrac a {\sin A} = \dfrac c {\sin C}$
Similarly, constructing the altitude from $A$ gives:
:$\dfrac b {\sin B} = \dfr... | Let $\triangle ABC$ be a [[Definition:Triangle (Geometry)|triangle]] whose sides $a, b, c$ are such that $a$ is opposite $A$, $b$ is opposite $B$ and $c$ is opposite $C$.
Then:
:$\dfrac a {\sin A} = \dfrac b {\sin B} = \dfrac c {\sin C} = 2 R$
where $R$ is the [[Definition:Circumradius of Triangle|circumradius]] of $... | Construct the [[Definition:Altitude of Triangle|altitude]] from $B$.
:[[File:Law Of Sines 1.png|240px]]
From the definition of [[Definition:Sine of Angle|sine]]:
:$\sin A = \dfrac h c$ and $\sin C = \dfrac h a$
Thus:
:$h = c \sin A$
and:
:$h = a \sin C$
This gives:
:$c \sin A = a \sin C$
So:
:$\dfrac a {\sin A} = ... | Law of Sines/Proof 1 | https://proofwiki.org/wiki/Law_of_Sines | https://proofwiki.org/wiki/Law_of_Sines/Proof_1 | [
"Law of Sines",
"Triangles",
"Sine Function",
"Named Theorems"
] | [
"Definition:Triangle (Geometry)",
"Definition:Circumcircle of Triangle/Circumradius"
] | [
"Definition:Altitude of Triangle",
"File:Law Of Sines 1.png",
"Definition:Sine/Definition from Triangle"
] |
proofwiki-1540 | Law of Sines | Let $\triangle ABC$ be a triangle whose sides $a, b, c$ are such that $a$ is opposite $A$, $b$ is opposite $B$ and $c$ is opposite $C$.
Then:
:$\dfrac a {\sin A} = \dfrac b {\sin B} = \dfrac c {\sin C} = 2 R$
where $R$ is the circumradius of $\triangle ABC$. | Construct the circumcircle of $\triangle ABC$, let $O$ be the circumcenter and $R$ be the circumradius.
Construct $\triangle AOB$ and let $E$ be the foot of the altitude of $\triangle AOB$ from $O$.
:350px
By the Inscribed Angle Theorem:
:$\angle ACB = \dfrac {\angle AOB} 2$
From the definition of the circumcenter:
:$A... | Let $\triangle ABC$ be a [[Definition:Triangle (Geometry)|triangle]] whose sides $a, b, c$ are such that $a$ is opposite $A$, $b$ is opposite $B$ and $c$ is opposite $C$.
Then:
:$\dfrac a {\sin A} = \dfrac b {\sin B} = \dfrac c {\sin C} = 2 R$
where $R$ is the [[Definition:Circumradius of Triangle|circumradius]] of $... | Construct the [[Definition:Circumcircle of Triangle|circumcircle]] of $\triangle ABC$, let $O$ be the [[Definition:Circumcenter of Triangle|circumcenter]] and $R$ be the [[Definition:Circumradius of Triangle|circumradius]].
Construct $\triangle AOB$ and let $E$ be the foot of the [[Definition:Altitude of Triangle|alti... | Law of Sines/Proof 2 | https://proofwiki.org/wiki/Law_of_Sines | https://proofwiki.org/wiki/Law_of_Sines/Proof_2 | [
"Law of Sines",
"Triangles",
"Sine Function",
"Named Theorems"
] | [
"Definition:Triangle (Geometry)",
"Definition:Circumcircle of Triangle/Circumradius"
] | [
"Definition:Circumcircle of Triangle",
"Definition:Circumcircle of Triangle/Circumcenter",
"Definition:Circumcircle of Triangle/Circumradius",
"Definition:Altitude of Triangle",
"File:Law-of-sines.png",
"Inscribed Angle Theorem",
"Definition:Circumcircle of Triangle/Circumcenter",
"Definition:Altitude... |
proofwiki-1541 | Law of Sines | Let $\triangle ABC$ be a triangle whose sides $a, b, c$ are such that $a$ is opposite $A$, $b$ is opposite $B$ and $c$ is opposite $C$.
Then:
:$\dfrac a {\sin A} = \dfrac b {\sin B} = \dfrac c {\sin C} = 2 R$
where $R$ is the circumradius of $\triangle ABC$. | === Acute Case ===
Let $\triangle ABC$ be acute.
:400px
Construct the circumcircle of $\triangle ABC$.
Let its radius be $R$.
Construct its diameter $BX$ through $B$.
By Thales' Theorem, $\angle BAX$ is a right angle.
From Angles in Same Segment of Circle are Equal:
:$\angle AXB = \angle ACB$
Then:
{{begin-eqn}}
{{eqn ... | Let $\triangle ABC$ be a [[Definition:Triangle (Geometry)|triangle]] whose sides $a, b, c$ are such that $a$ is opposite $A$, $b$ is opposite $B$ and $c$ is opposite $C$.
Then:
:$\dfrac a {\sin A} = \dfrac b {\sin B} = \dfrac c {\sin C} = 2 R$
where $R$ is the [[Definition:Circumradius of Triangle|circumradius]] of $... | === Acute Case ===
Let $\triangle ABC$ be [[Definition:Acute Triangle|acute]].
:[[File:Law-of-sines-proof-3.png|400px]]
Construct the [[Definition:Circumcircle of Triangle|circumcircle]] of $\triangle ABC$.
Let its [[Definition:Radius of Circle|radius]] be $R$.
Construct its [[Definition:Diameter of Circle|diamete... | Law of Sines/Proof 3 | https://proofwiki.org/wiki/Law_of_Sines | https://proofwiki.org/wiki/Law_of_Sines/Proof_3 | [
"Law of Sines",
"Triangles",
"Sine Function",
"Named Theorems"
] | [
"Definition:Triangle (Geometry)",
"Definition:Circumcircle of Triangle/Circumradius"
] | [
"Definition:Triangle (Geometry)/Acute",
"File:Law-of-sines-proof-3.png",
"Definition:Circumcircle of Triangle",
"Definition:Circle/Radius",
"Definition:Circle/Diameter",
"Thales' Theorem",
"Definition:Right Angle",
"Angles in Same Segment of Circle are Equal",
"Definition:Polygon/Vertex",
"Definit... |
proofwiki-1542 | Law of Sines | Let $\triangle ABC$ be a triangle whose sides $a, b, c$ are such that $a$ is opposite $A$, $b$ is opposite $B$ and $c$ is opposite $C$.
Then:
:$\dfrac a {\sin A} = \dfrac b {\sin B} = \dfrac c {\sin C} = 2 R$
where $R$ is the circumradius of $\triangle ABC$. | {{begin-eqn}}
{{eqn | l = \sin b \sin c \cos A
| r = \cos a - \cos b \cos c
| c = Spherical Law of Cosines
}}
{{eqn | ll= \leadsto
| l = \sin^2 b \sin^2 c \cos^2 A
| r = \cos^2 a - 2 \cos a \cos b \cos c + \cos^2 b \cos^2 c
| c =
}}
{{eqn | ll= \leadsto
| l = \sin^2 b \sin^2 c \pare... | Let $\triangle ABC$ be a [[Definition:Triangle (Geometry)|triangle]] whose sides $a, b, c$ are such that $a$ is opposite $A$, $b$ is opposite $B$ and $c$ is opposite $C$.
Then:
:$\dfrac a {\sin A} = \dfrac b {\sin B} = \dfrac c {\sin C} = 2 R$
where $R$ is the [[Definition:Circumradius of Triangle|circumradius]] of $... | {{begin-eqn}}
{{eqn | l = \sin b \sin c \cos A
| r = \cos a - \cos b \cos c
| c = [[Spherical Law of Cosines]]
}}
{{eqn | ll= \leadsto
| l = \sin^2 b \sin^2 c \cos^2 A
| r = \cos^2 a - 2 \cos a \cos b \cos c + \cos^2 b \cos^2 c
| c =
}}
{{eqn | ll= \leadsto
| l = \sin^2 b \sin^2 c \... | Spherical Law of Sines/Proof 1 | https://proofwiki.org/wiki/Law_of_Sines | https://proofwiki.org/wiki/Spherical_Law_of_Sines/Proof_1 | [
"Law of Sines",
"Triangles",
"Sine Function",
"Named Theorems"
] | [
"Definition:Triangle (Geometry)",
"Definition:Circumcircle of Triangle/Circumradius"
] | [
"Spherical Law of Cosines",
"Sum of Squares of Sine and Cosine",
"Sum of Squares of Sine and Cosine",
"Definition:Spherical Triangle",
"Definition:Spherical Triangle/Side",
"Definition:Angular Measure/Radian",
"Definition:Spherical Angle",
"Shape of Sine Function",
"Definition:Square Root/Negative",... |
proofwiki-1543 | Law of Sines | Let $\triangle ABC$ be a triangle whose sides $a, b, c$ are such that $a$ is opposite $A$, $b$ is opposite $B$ and $c$ is opposite $C$.
Then:
:$\dfrac a {\sin A} = \dfrac b {\sin B} = \dfrac c {\sin C} = 2 R$
where $R$ is the circumradius of $\triangle ABC$. | :500px
Let $A$, $B$ and $C$ be the vertices of a spherical triangle on the surface of a sphere $S$.
By definition of a spherical triangle, $AB$, $BC$ and $AC$ are arcs of great circles on $S$.
By definition of a great circle, the center of each of these great circles is $O$.
Let $O$ be joined to each of $A$, $B$ and $C... | Let $\triangle ABC$ be a [[Definition:Triangle (Geometry)|triangle]] whose sides $a, b, c$ are such that $a$ is opposite $A$, $b$ is opposite $B$ and $c$ is opposite $C$.
Then:
:$\dfrac a {\sin A} = \dfrac b {\sin B} = \dfrac c {\sin C} = 2 R$
where $R$ is the [[Definition:Circumradius of Triangle|circumradius]] of $... | :[[File:Spherical-Cosine-Formula-2.png|500px]]
Let $A$, $B$ and $C$ be the [[Definition:Vertex of Polygon|vertices]] of a [[Definition:Spherical Triangle|spherical triangle]] on the surface of a [[Definition:Sphere (Geometry)|sphere]] $S$.
By definition of a [[Definition:Spherical Triangle|spherical triangle]], $AB$,... | Spherical Law of Sines/Proof 2 | https://proofwiki.org/wiki/Law_of_Sines | https://proofwiki.org/wiki/Spherical_Law_of_Sines/Proof_2 | [
"Law of Sines",
"Triangles",
"Sine Function",
"Named Theorems"
] | [
"Definition:Triangle (Geometry)",
"Definition:Circumcircle of Triangle/Circumradius"
] | [
"File:Spherical-Cosine-Formula-2.png",
"Definition:Polygon/Vertex",
"Definition:Spherical Triangle",
"Definition:Sphere/Geometry",
"Definition:Spherical Triangle",
"Definition:Circle/Arc",
"Definition:Great Circle",
"Definition:Great Circle",
"Definition:Circle/Center",
"Definition:Great Circle",
... |
proofwiki-1544 | Negative of Supremum is Infimum of Negatives | Let $S$ be a non-empty subset of the real numbers $\R$.
Let $S$ be bounded above.
Then:
:$(1): \quad \set {x \in \R: -x \in S}$ is bounded below
:$(2): \quad \ds -\sup_{x \mathop \in S} x = \map {\inf_{x \mathop \in S} } {-x}$
where $\sup$ and $\inf$ denote the supremum and infimum respectively. | As $S$ is non-empty and bounded above, it follows by the Continuum Property that $S$ admits a supremum.
Let $B = \sup S$.
Let $T = \set {x \in \R: -x \in S}$.
By definition, $B$ is an upper bound for $S$.
From Negative of Upper Bound of Set of Real Numbers is Lower Bound of Negatives:
:$-B$ is a lower bound for $T$.
Th... | Let $S$ be a [[Definition:Non-Empty Set|non-empty]] [[Definition:Subset|subset]] of the [[Definition:Real Number|real numbers]] $\R$.
Let $S$ be [[Definition:Bounded Above Subset of Real Numbers|bounded above]].
Then:
:$(1): \quad \set {x \in \R: -x \in S}$ is [[Definition:Bounded Below Subset of Real Numbers|bounde... | As $S$ is [[Definition:Non-Empty Set|non-empty]] and [[Definition:Bounded Above Subset of Real Numbers|bounded above]], it follows by the [[Continuum Property]] that $S$ admits a [[Definition:Supremum of Subset of Real Numbers|supremum]].
Let $B = \sup S$.
Let $T = \set {x \in \R: -x \in S}$.
By definition, $B$ is ... | Negative of Supremum is Infimum of Negatives | https://proofwiki.org/wiki/Negative_of_Supremum_is_Infimum_of_Negatives | https://proofwiki.org/wiki/Negative_of_Supremum_is_Infimum_of_Negatives | [
"Real Analysis"
] | [
"Definition:Non-Empty Set",
"Definition:Subset",
"Definition:Real Number",
"Definition:Bounded Above Set/Real Numbers",
"Definition:Bounded Below Set/Real Numbers",
"Definition:Supremum of Set/Real Numbers",
"Definition:Infimum of Set/Real Numbers"
] | [
"Definition:Non-Empty Set",
"Definition:Bounded Above Set/Real Numbers",
"Continuum Property",
"Definition:Supremum of Set/Real Numbers",
"Definition:Upper Bound of Set/Real Numbers",
"Negative of Upper Bound of Set of Real Numbers is Lower Bound of Negatives",
"Definition:Lower Bound of Set/Real Number... |
proofwiki-1545 | Negative of Infimum is Supremum of Negatives | Let $T$ be a non-empty subset of the real numbers $\R$.
Let $T$ be bounded below.
Then:
:$(1): \quad \set {x \in \R: -x \in T}$ is bounded above
:$(2): \quad \ds -\inf_{x \mathop \in T} x = \map {\sup_{x \mathop \in T} } {-x}$
where $\sup$ and $\inf$ denote the supremum and infimum respectively. | As $T$ is non-empty and bounded below, it follows by the Continuum Property that $T$ admits an infimum.
Let $B = \inf T$.
Let $S = \set {x \in \R: -x \in T}$.
Since $\forall x \in T: x \ge B$ it follows that:
:$\forall x \in T: -x \le -B$
Hence $-B$ is an upper bound for $S$, and so $\set {x \in \R: -x \in T}$ is bound... | Let $T$ be a [[Definition:Non-Empty Set|non-empty]] [[Definition:Subset|subset]] of the [[Definition:Real Number|real numbers]] $\R$.
Let $T$ be [[Definition:Bounded Below Subset of Real Numbers|bounded below]].
Then:
:$(1): \quad \set {x \in \R: -x \in T}$ is [[Definition:Bounded Above Subset of Real Numbers|bounde... | As $T$ is [[Definition:Non-Empty Set|non-empty]] and [[Definition:Bounded Below Set|bounded below]], it follows by the [[Continuum Property]] that $T$ admits an [[Definition:Infimum of Subset of Real Numbers|infimum]].
Let $B = \inf T$.
Let $S = \set {x \in \R: -x \in T}$.
Since $\forall x \in T: x \ge B$ it follow... | Negative of Infimum is Supremum of Negatives | https://proofwiki.org/wiki/Negative_of_Infimum_is_Supremum_of_Negatives | https://proofwiki.org/wiki/Negative_of_Infimum_is_Supremum_of_Negatives | [
"Real Analysis"
] | [
"Definition:Non-Empty Set",
"Definition:Subset",
"Definition:Real Number",
"Definition:Bounded Below Set/Real Numbers",
"Definition:Bounded Above Set/Real Numbers",
"Definition:Supremum of Set/Real Numbers",
"Definition:Infimum of Set/Real Numbers"
] | [
"Definition:Non-Empty Set",
"Definition:Bounded Below Set",
"Continuum Property",
"Definition:Infimum of Set/Real Numbers",
"Definition:Upper Bound of Set/Real Numbers",
"Definition:Bounded Above Set/Real Numbers",
"Definition:Supremum of Set/Real Numbers",
"Definition:Supremum of Set/Real Numbers",
... |
proofwiki-1546 | Infimum Plus Constant | Let $T$ be a subset of the set of real numbers.
Let $T$ be bounded below.
Let $\xi \in \R$.
Then:
:$\ds \map {\inf_{x \mathop \in T} } {x + \xi} = \xi + \map {\inf_{x \mathop \in T} } x$
where $\inf$ denotes infimum. | From Negative of Infimum is Supremum of Negatives, we have that:
:$\ds -\inf_{x \mathop \in T} x = \map {\sup_{x \mathop \in T} } {-x} \implies \inf_{x \mathop \in T} x = -\map {\sup_{x \mathop \in T} } {-x}$
Let $S = \set {x \in \R: -x \in T}$.
From Negative of Infimum is Supremum of Negatives, $S$ is bounded above.
W... | Let $T$ be a [[Definition:Subset|subset]] of the set of [[Definition:Real Number|real numbers]].
Let $T$ be [[Definition:Bounded Below Set|bounded below]].
Let $\xi \in \R$.
Then:
:$\ds \map {\inf_{x \mathop \in T} } {x + \xi} = \xi + \map {\inf_{x \mathop \in T} } x$
where $\inf$ denotes [[Definition:Infimum of Se... | From [[Negative of Infimum is Supremum of Negatives]], we have that:
:$\ds -\inf_{x \mathop \in T} x = \map {\sup_{x \mathop \in T} } {-x} \implies \inf_{x \mathop \in T} x = -\map {\sup_{x \mathop \in T} } {-x}$
Let $S = \set {x \in \R: -x \in T}$.
From [[Negative of Infimum is Supremum of Negatives]], $S$ is [[Defi... | Infimum Plus Constant | https://proofwiki.org/wiki/Infimum_Plus_Constant | https://proofwiki.org/wiki/Infimum_Plus_Constant | [
"Real Analysis"
] | [
"Definition:Subset",
"Definition:Real Number",
"Definition:Bounded Below Set",
"Definition:Infimum of Set"
] | [
"Negative of Infimum is Supremum of Negatives",
"Negative of Infimum is Supremum of Negatives",
"Definition:Bounded Above Set",
"Supremum Plus Constant"
] |
proofwiki-1547 | Infimum of Subset | Let $\struct {U, \preceq}$ be an ordered set.
Let $S \subseteq U$.
Let $T \subseteq S$.
Let $\struct {S, \preceq}$ admit an infimum in $U$.
If $T$ also admits an infimum in $U$, then $\map \inf S \preceq \map \inf T$. | Let $B = \map \inf S$.
Then $B$ is a lower bound for $S$.
As $T \subseteq S$, it follows by the definition of a subset that $x \in T \implies x \in S$.
Because $x \in S \implies B \preceq x$ (as $B$ is a lower bound for $S$) it follows that $x \in T \implies B \preceq x$.
So $B$ is a lower bound for $T$.
Therefore $B$ ... | Let $\struct {U, \preceq}$ be an [[Definition:Ordered Set|ordered set]].
Let $S \subseteq U$.
Let $T \subseteq S$.
Let $\struct {S, \preceq}$ admit an [[Definition:Infimum of Set|infimum]] in $U$.
If $T$ also admits an [[Definition:Infimum of Set|infimum]] in $U$, then $\map \inf S \preceq \map \inf T$. | Let $B = \map \inf S$.
Then $B$ is a [[Definition:Lower Bound of Set|lower bound]] for $S$.
As $T \subseteq S$, it follows by the definition of a [[Definition:Subset|subset]] that $x \in T \implies x \in S$.
Because $x \in S \implies B \preceq x$ (as $B$ is a [[Definition:Lower Bound of Set|lower bound]] for $S$) it... | Infimum of Subset | https://proofwiki.org/wiki/Infimum_of_Subset | https://proofwiki.org/wiki/Infimum_of_Subset | [
"Order Theory"
] | [
"Definition:Ordered Set",
"Definition:Infimum of Set",
"Definition:Infimum of Set"
] | [
"Definition:Lower Bound of Set",
"Definition:Subset",
"Definition:Lower Bound of Set",
"Definition:Lower Bound of Set",
"Definition:Ordering",
"Definition:Infimum of Set"
] |
proofwiki-1548 | Construction of Equilateral Triangle | On a given straight line segment, it is possible to construct an equilateral triangle.
{{:Euclid:Proposition/I/1}} | As $A$ is the center of circle $BCD$, it follows from {{EuclidDefLink|I|15|Circle}} that $AC = AB$.
As $B$ is the center of circle $ACE$, it follows from {{EuclidDefLink|I|15|Circle}} that $BC = AB$.
So, as $AC = AB$ and $BC = AB$, it follows from {{EuclidCommonNotionLink|1}} that $AC = BC$.
Therefore $AB = AC = BC$.
T... | On a given [[Definition:Line Segment|straight line segment]], it is possible to construct an [[Definition:Equilateral Triangle|equilateral triangle]].
{{:Euclid:Proposition/I/1}} | As $A$ is the [[Definition:Center of Circle|center]] of circle $BCD$, it follows from {{EuclidDefLink|I|15|Circle}} that $AC = AB$.
As $B$ is the [[Definition:Center of Circle|center]] of circle $ACE$, it follows from {{EuclidDefLink|I|15|Circle}} that $BC = AB$.
So, as $AC = AB$ and $BC = AB$, it follows from {{Euc... | Construction of Equilateral Triangle | https://proofwiki.org/wiki/Construction_of_Equilateral_Triangle | https://proofwiki.org/wiki/Construction_of_Equilateral_Triangle | [
"Equilateral Triangles",
"Geometric Constructions"
] | [
"Definition:Line/Segment",
"Definition:Triangle (Geometry)/Equilateral"
] | [
"Definition:Circle/Center",
"Definition:Circle/Center",
"Definition:Triangle (Geometry)/Equilateral"
] |
proofwiki-1549 | Construction of Equal Straight Line | At a given point, it is possible to construct a straight line segment of length equal to that of any given straight line segment.
The given point will be an endpoint of the constructed straight line segment.
{{:Euclid:Proposition/I/2}} | As $B$ is the center of circle $CGH$, it follows from {{EuclidDefLink|I|15|Circle}} that $BC = BG$.
As $D$ is the center of circle $GKL$, it follows from {{EuclidDefLink|I|15|Circle}} that $DL = DG$.
As $\triangle ABD$ is an equilateral triangle, it follows that $DA = DB$.
Therefore, by {{EuclidCommonNotionLink|3}}, $A... | At a given [[Definition:Point|point]], it is possible to construct a [[Definition:Line Segment|straight line segment]] of [[Definition:Length (Linear Measure)|length]] equal to that of any given [[Definition:Line Segment|straight line segment]].
The given [[Definition:Point|point]] will be an [[Definition:Endpoint of ... | As $B$ is the [[Definition:Center of Circle|center]] of circle $CGH$, it follows from {{EuclidDefLink|I|15|Circle}} that $BC = BG$.
As $D$ is the [[Definition:Center of Circle|center]] of circle $GKL$, it follows from {{EuclidDefLink|I|15|Circle}} that $DL = DG$.
As $\triangle ABD$ is an [[Definition:Equilateral Tria... | Construction of Equal Straight Line | https://proofwiki.org/wiki/Construction_of_Equal_Straight_Line | https://proofwiki.org/wiki/Construction_of_Equal_Straight_Line | [
"Lines"
] | [
"Definition:Point",
"Definition:Line/Segment",
"Definition:Linear Measure/Length",
"Definition:Line/Segment",
"Definition:Point",
"Definition:Line/Endpoint",
"Definition:Line/Segment"
] | [
"Definition:Circle/Center",
"Definition:Circle/Center",
"Definition:Triangle (Geometry)/Equilateral",
"Definition:Line/Segment"
] |
proofwiki-1550 | Construction of Equal Straight Lines from Unequal | Given two unequal straight line segments, it is possible to cut off from the greater a straight line segment equal to the lesser.
{{:Euclid:Proposition/I/3}} | As $A$ is the center of circle $DEF$, it follows from {{EuclidDefLink|I|15|Circle}} that $AE = AD$.
But $C$ is also equal to $AD$.
So, as $C = AD$ and $AD = AE$, it follows from Common Notion 1 that $AE = C$.
Therefore, given the two straight line segments $AB$ and $C$, from the greater of these $AB$, a length $AE$ has... | Given two unequal [[Definition:Line Segment|straight line segments]], it is possible to cut off from the greater a [[Definition:Line Segment|straight line segment]] equal to the lesser.
{{:Euclid:Proposition/I/3}} | As $A$ is the [[Definition:Center of Circle|center]] of circle $DEF$, it follows from {{EuclidDefLink|I|15|Circle}} that $AE = AD$.
But $C$ is also equal to $AD$.
So, as $C = AD$ and $AD = AE$, it follows from [[Axiom:Euclid's Common Notions|Common Notion 1]] that $AE = C$.
Therefore, given the two [[Definition:Lin... | Construction of Equal Straight Lines from Unequal | https://proofwiki.org/wiki/Construction_of_Equal_Straight_Lines_from_Unequal | https://proofwiki.org/wiki/Construction_of_Equal_Straight_Lines_from_Unequal | [
"Lines"
] | [
"Definition:Line/Segment",
"Definition:Line/Segment"
] | [
"Definition:Circle/Center",
"Axiom:Euclid's Common Notions",
"Definition:Line/Segment"
] |
proofwiki-1551 | Triangle Side-Angle-Side Congruence | If $2$ triangles have:
:$2$ sides equal to $2$ sides respectively
:the angles contained by the equal straight lines equal
they will also have:
:their third sides equal
:the remaining two angles equal to their respective remaining angles, namely, those which the equal sides subtend. | 500px
Let $\triangle ABC$ and $\triangle DEF$ be $2$ triangles having sides $AB = DE$ and $AC = DF$, and with $\angle BAC = \angle EDF$.
If $\triangle ABC$ is placed on $\triangle DEF$ such that:
:the point $A$ is placed on point $D$, and
:the line $AB$ is placed on line $DE$
then the point $B$ will also coincide with... | If $2$ [[Definition:Triangle (Geometry)|triangles]] have:
:$2$ [[Definition:Side of Polygon|sides]] equal to $2$ [[Definition:Side of Polygon|sides]] respectively
:the [[Definition:Angle|angles]] [[Definition:Containment of Angle|contained]] by the equal [[Definition:Straight Line Segment|straight lines]] equal
they w... | [[File:Euclid-I-4.png|500px]]
Let $\triangle ABC$ and $\triangle DEF$ be $2$ [[Definition:Triangle (Geometry)|triangles]] having [[Definition:Side of Polygon|sides]] $AB = DE$ and $AC = DF$, and with $\angle BAC = \angle EDF$.
If $\triangle ABC$ is placed on $\triangle DEF$ such that:
:the [[Definition:Point|point]] ... | Triangle Side-Angle-Side Congruence | https://proofwiki.org/wiki/Triangle_Side-Angle-Side_Congruence | https://proofwiki.org/wiki/Triangle_Side-Angle-Side_Congruence | [
"Triangle Side-Angle-Side Congruence",
"Triangles",
"Congruence (Geometry)"
] | [
"Definition:Triangle (Geometry)",
"Definition:Polygon/Side",
"Definition:Polygon/Side",
"Definition:Angle",
"Definition:Angle/Containment",
"Definition:Line/Straight Line Segment",
"Definition:Polygon/Side",
"Definition:Angle",
"Definition:Angle",
"Definition:Polygon/Side",
"Definition:Subtend"
... | [
"File:Euclid-I-4.png",
"Definition:Triangle (Geometry)",
"Definition:Polygon/Side",
"Definition:Point",
"Definition:Point",
"Definition:Line/Straight Line Segment",
"Definition:Line/Straight Line Segment",
"Definition:Point",
"Definition:Point",
"Definition:Line/Straight Line Segment",
"Definiti... |
proofwiki-1552 | Isosceles Triangle has Two Equal Angles | In isosceles triangles, the angles at the base are equal to each other.
Also, if the equal straight lines are extended, the angles under the base will also be equal to each other.
{{:Euclid:Proposition/I/5}} | 200px
Let $\triangle ABC$ be an isosceles triangle whose side $AB$ equals side $AC$.
We extend the straight lines $AB$ and $AC$ to $D$ and $E$ respectively.
Let $F$ be a point on $BD$.
We cut off from $AE$ a length $AG$ equal to $AF$.
We draw line segments $FC$ and $GB$.
Since $AF = AG$ and $AB = AC$, the two sides $FA... | In [[Definition:Isosceles Triangle|isosceles triangles]], the [[Definition:Angle|angles]] at the [[Definition:Base of Isosceles Triangle|base]] are equal to each other.
Also, if the equal [[Definition:Line Segment|straight lines]] are extended, the [[Definition:Angle|angles]] under the [[Definition:Base of Isosceles T... | [[File:Euclid-I-5.png|200px]]
Let $\triangle ABC$ be an [[Definition:Isosceles Triangle|isosceles triangle]] whose side $AB$ equals side $AC$.
We [[Axiom:Euclid's Second Postulate|extend the straight lines]] $AB$ and $AC$ to $D$ and $E$ respectively.
Let $F$ be a point on $BD$.
We [[Construction of Equal Straight ... | Isosceles Triangle has Two Equal Angles | https://proofwiki.org/wiki/Isosceles_Triangle_has_Two_Equal_Angles | https://proofwiki.org/wiki/Isosceles_Triangle_has_Two_Equal_Angles | [
"Isosceles Triangle has Two Equal Angles",
"Isosceles Triangles"
] | [
"Definition:Triangle (Geometry)/Isosceles",
"Definition:Angle",
"Definition:Triangle (Geometry)/Isosceles/Base",
"Definition:Line/Segment",
"Definition:Angle",
"Definition:Triangle (Geometry)/Isosceles/Base"
] | [
"File:Euclid-I-5.png",
"Definition:Triangle (Geometry)/Isosceles",
"Axiom:Euclid's Second Postulate",
"Construction of Equal Straight Lines from Unequal",
"Axiom:Euclid's First Postulate",
"Definition:Angle/Containment",
"Definition:Angle",
"Triangle Side-Angle-Side Congruence",
"Triangle Side-Angle... |
proofwiki-1553 | Triangle with Two Equal Angles is Isosceles | If a triangle has two angles equal to each other, the sides which subtend the equal angles will also be equal to one another.
Hence, by definition, such a triangle will be isosceles.
{{:Euclid:Proposition/I/6}} | :200px
Let $\triangle ABC$ be a triangle in which $\angle ABC = \angle ACB$.
Suppose side $AB$ is not equal to side $AC$. Then one of them will be greater.
{{WLOG}}, Suppose $AB > AC$.
We cut off from $AB$ a length $DB$ equal to $AC$.
We draw the line segment $CD$.
Since $DB = AC$, and $BC$ is common, the two sides $DB... | If a [[Definition:Triangle (Geometry)|triangle]] has two [[Definition:Angle|angles]] equal to each other, the [[Definition:Side of Polygon|sides]] which [[Definition:Subtend|subtend]] the equal [[Definition:Angle|angles]] will also be equal to one another.
Hence, by definition, such a triangle will be [[Definition:Iso... | :[[File:Euclid-I-6.png|200px]]
Let $\triangle ABC$ be a [[Definition:Triangle (Geometry)|triangle]] in which $\angle ABC = \angle ACB$.
Suppose side $AB$ is not equal to side $AC$. Then one of them will be greater.
{{WLOG}}, Suppose $AB > AC$.
We [[Construction of Equal Straight Lines from Unequal|cut off from $AB... | Triangle with Two Equal Angles is Isosceles/Proof 1 | https://proofwiki.org/wiki/Triangle_with_Two_Equal_Angles_is_Isosceles | https://proofwiki.org/wiki/Triangle_with_Two_Equal_Angles_is_Isosceles/Proof_1 | [
"Isosceles Triangles",
"Triangle with Two Equal Angles is Isosceles"
] | [
"Definition:Triangle (Geometry)",
"Definition:Angle",
"Definition:Polygon/Side",
"Definition:Subtend",
"Definition:Angle",
"Definition:Triangle (Geometry)/Isosceles"
] | [
"File:Euclid-I-6.png",
"Definition:Triangle (Geometry)",
"Construction of Equal Straight Lines from Unequal",
"Axiom:Euclid's First Postulate",
"Triangle Side-Angle-Side Congruence"
] |
proofwiki-1554 | Triangle with Two Equal Angles is Isosceles | If a triangle has two angles equal to each other, the sides which subtend the equal angles will also be equal to one another.
Hence, by definition, such a triangle will be isosceles.
{{:Euclid:Proposition/I/6}} | Let $\angle ABC$ and $\angle ACB$ be the angles that are the same.
{{begin-eqn}}
{{eqn | n = 1
| l = \angle ABC
| r = \angle ACB
| c = {{hypothesis}}
}}
{{eqn | n = 2
| l = BC
| r = CB
| c = Equality is Reflexive
}}
{{eqn | n = 3
| l = \angle ACB
| r = \angle ABC
| ... | If a [[Definition:Triangle (Geometry)|triangle]] has two [[Definition:Angle|angles]] equal to each other, the [[Definition:Side of Polygon|sides]] which [[Definition:Subtend|subtend]] the equal [[Definition:Angle|angles]] will also be equal to one another.
Hence, by definition, such a triangle will be [[Definition:Iso... | Let $\angle ABC$ and $\angle ACB$ be the [[Definition:Angle|angles]] that are the same.
{{begin-eqn}}
{{eqn | n = 1
| l = \angle ABC
| r = \angle ACB
| c = {{hypothesis}}
}}
{{eqn | n = 2
| l = BC
| r = CB
| c = [[Equality is Reflexive]]
}}
{{eqn | n = 3
| l = \angle ACB
... | Triangle with Two Equal Angles is Isosceles/Proof 2 | https://proofwiki.org/wiki/Triangle_with_Two_Equal_Angles_is_Isosceles | https://proofwiki.org/wiki/Triangle_with_Two_Equal_Angles_is_Isosceles/Proof_2 | [
"Isosceles Triangles",
"Triangle with Two Equal Angles is Isosceles"
] | [
"Definition:Triangle (Geometry)",
"Definition:Angle",
"Definition:Polygon/Side",
"Definition:Subtend",
"Definition:Angle",
"Definition:Triangle (Geometry)/Isosceles"
] | [
"Definition:Angle",
"Equality is Reflexive",
"Triangle Angle-Side-Angle Congruence"
] |
proofwiki-1555 | Two Lines Meet at Unique Point | Let two straight line segments be constructed on a straight line segment from its endpoints so that they meet at a point.
Then there cannot be two other straight line segments equal to the former two respectively, constructed on the same straight line segment and on the same side of it, meeting at a different point.
{{... | :400px
Let $AC$ and $CB$ be constructed on $AB$ meeting at $C$.
Let two other straight line segments $AD$ and $DB$ be constructed on $AB$, on the same side of it, meeting at $D$, such that $AC = AD$ and $CB = DB$.
{{AimForCont}} $C$ and $D$ are different points.
Let $CD$ be joined.
We have {{hypothesis}} that:
:$AC = A... | Let two [[Definition:Line Segment|straight line segments]] be constructed on a [[Definition:Line Segment|straight line segment]] from its [[Definition:Endpoint of Line|endpoints]] so that they meet at a [[Definition:Point|point]].
Then there cannot be two other [[Definition:Line Segment|straight line segments]] equal ... | :[[File:Euclid-I-7.png|400px]]
Let $AC$ and $CB$ be constructed on $AB$ meeting at $C$.
Let two other [[Definition:Line Segment|straight line segments]] $AD$ and $DB$ be constructed on $AB$, on the same side of it, meeting at $D$, such that $AC = AD$ and $CB = DB$.
{{AimForCont}} $C$ and $D$ are different [[Definit... | Two Lines Meet at Unique Point | https://proofwiki.org/wiki/Two_Lines_Meet_at_Unique_Point | https://proofwiki.org/wiki/Two_Lines_Meet_at_Unique_Point | [
"Lines"
] | [
"Definition:Line/Segment",
"Definition:Line/Segment",
"Definition:Line/Endpoint",
"Definition:Point",
"Definition:Line/Segment",
"Definition:Line/Segment",
"Definition:Point"
] | [
"File:Euclid-I-7.png",
"Definition:Line/Segment",
"Definition:Point",
"Axiom:Euclid's Common Notions",
"Definition:Contradiction",
"Proof by Contradiction",
"Definition:Point"
] |
proofwiki-1556 | Triangle Side-Side-Side Congruence | Let two triangles have all $3$ sides equal.
Then they also have all $3$ angles equal.
Thus two triangles whose sides are all equal are themselves congruent.
{{:Euclid:Proposition/I/8}} | :500px
Let $\triangle ABC$ and $\triangle DEF$ be two triangles such that:
:$AB = DE$
:$AC = DF$
:$BC = EF$
Suppose $\triangle ABC$ were superimposed over $\triangle DEF$ so that point $B$ is placed on point $E$ and the side $BC$ on $EF$.
Then $C$ will coincide with $F$, as $BC = EF$ and so $BC$ coincides with $EF$.
{{... | Let two [[Definition:Triangle (Geometry)|triangles]] have all $3$ [[Definition:Side of Polygon|sides]] equal.
Then they also have all $3$ [[Definition:Angle|angles]] equal.
Thus two [[Definition:Triangle (Geometry)|triangles]] whose [[Definition:Side of Polygon|sides]] are all equal are themselves [[Definition:Congr... | :[[File:Euclid-I-8.png|500px]]
Let $\triangle ABC$ and $\triangle DEF$ be two [[Definition:Triangle (Geometry)|triangles]] such that:
:$AB = DE$
:$AC = DF$
:$BC = EF$
Suppose $\triangle ABC$ were superimposed over $\triangle DEF$ so that point $B$ is placed on point $E$ and the [[Definition:Side of Polygon|side]] $BC... | Triangle Side-Side-Side Congruence/Proof 1 | https://proofwiki.org/wiki/Triangle_Side-Side-Side_Congruence | https://proofwiki.org/wiki/Triangle_Side-Side-Side_Congruence/Proof_1 | [
"Triangle Side-Side-Side Congruence",
"Triangles",
"Congruence (Geometry)"
] | [
"Definition:Triangle (Geometry)",
"Definition:Polygon/Side",
"Definition:Angle",
"Definition:Triangle (Geometry)",
"Definition:Polygon/Side",
"Definition:Congruence (Geometry)"
] | [
"File:Euclid-I-8.png",
"Definition:Triangle (Geometry)",
"Definition:Polygon/Side",
"Definition:Line/Segment",
"Definition:Line/Segment",
"Definition:Point",
"Definition:Point",
"Proof by Contradiction",
"Definition:Polygon/Side",
"Definition:Angle"
] |
proofwiki-1557 | Triangle Side-Side-Side Congruence | Let two triangles have all $3$ sides equal.
Then they also have all $3$ angles equal.
Thus two triangles whose sides are all equal are themselves congruent.
{{:Euclid:Proposition/I/8}} | Let $\triangle ABC$ and $\triangle DEF$ have all three pairs of corresponding sides equal.
Let $AC$ be the longest side of $\triangle ABC$.
If there is more than one longest side, choose an arbitrary one.
We have {{hypothesis}}:
:$AC = DF$
{{tidy}}
There are two cases:
'''case 1'''
The two triangles can be arranged a... | Let two [[Definition:Triangle (Geometry)|triangles]] have all $3$ [[Definition:Side of Polygon|sides]] equal.
Then they also have all $3$ [[Definition:Angle|angles]] equal.
Thus two [[Definition:Triangle (Geometry)|triangles]] whose [[Definition:Side of Polygon|sides]] are all equal are themselves [[Definition:Congr... | Let $\triangle ABC$ and $\triangle DEF$ have all three pairs of corresponding [[Definition:Side of Polygon|sides]] equal.
Let $AC$ be the longest [[Definition:Side of Polygon|side]] of $\triangle ABC$.
If there is more than one longest [[Definition:Side of Polygon|side]], choose an arbitrary one.
We have {{hypothe... | Triangle Side-Side-Side Congruence/Proof 2 | https://proofwiki.org/wiki/Triangle_Side-Side-Side_Congruence | https://proofwiki.org/wiki/Triangle_Side-Side-Side_Congruence/Proof_2 | [
"Triangle Side-Side-Side Congruence",
"Triangles",
"Congruence (Geometry)"
] | [
"Definition:Triangle (Geometry)",
"Definition:Polygon/Side",
"Definition:Angle",
"Definition:Triangle (Geometry)",
"Definition:Polygon/Side",
"Definition:Congruence (Geometry)"
] | [
"Definition:Polygon/Side",
"Definition:Polygon/Side",
"Definition:Polygon/Side",
"Definition:Triangle",
"Definition:Reflection (Geometry)",
"File:Side-Side-Side1.png",
"Triangle Side-Angle-Side Congruence",
"Mathematician:Euclid",
"Euclid:Proposition/I/4",
"Definition:Triangle",
"Definition:Tria... |
proofwiki-1558 | Bisection of Angle | It is possible to bisect any given rectilineal angle.
{{:Euclid:Proposition/I/9}} | We have:
:$AD = AE$
:$AF$ is common
:$DF = EF$
Thus from {{EuclidPropLink|prop = 8|title = Triangle Side-Side-Side Congruence}}:
:$\triangle ADF \sim \triangle AEF$
Thus:
:$\angle DAF = \angle EAF$
Hence $\angle BAC$ has been bisected by $AF$.
{{qed}}
{{Euclid Note|9|I|There are quicker and easier constructions of a bi... | It is possible to [[Definition:Bisection|bisect]] any given [[Definition:Rectilineal Angle|rectilineal angle]].
{{:Euclid:Proposition/I/9}} | We have:
:$AD = AE$
:$AF$ is common
:$DF = EF$
Thus from {{EuclidPropLink|prop = 8|title = Triangle Side-Side-Side Congruence}}:
:$\triangle ADF \sim \triangle AEF$
Thus:
:$\angle DAF = \angle EAF$
Hence $\angle BAC$ has been [[Definition:Bisection|bisected]] by $AF$.
{{qed}}
{{Euclid Note|9|I|There are quicker an... | Bisection of Angle | https://proofwiki.org/wiki/Bisection_of_Angle | https://proofwiki.org/wiki/Bisection_of_Angle | [
"Bisection of Angle",
"Angles",
"Angle Bisectors"
] | [
"Definition:Bisection",
"Definition:Angle/Rectilineal"
] | [
"Definition:Bisection"
] |
proofwiki-1559 | Bisection of Straight Line | It is possible to bisect a straight line segment.
{{:Euclid:Proposition/I/10}} | As $\triangle ABC$ is an equilateral triangle, it follows that $AC = CB$.
The two triangles $\triangle ACD$ and $\triangle BCD$ have side $CD$ in common, and side $AC$ of $\triangle ACD$ equals side $BC$ of $\triangle BCD$.
The angle $\angle ACD$ subtended by lines $AC$ and $CD$ equals the angle $\angle BCD$ subtended ... | It is possible to [[Definition:Bisection|bisect]] a [[Definition:Line Segment|straight line segment]].
{{:Euclid:Proposition/I/10}} | As $\triangle ABC$ is an [[Definition:Equilateral Triangle|equilateral triangle]], it follows that $AC = CB$.
The two [[Definition:Triangle (Geometry)|triangles]] $\triangle ACD$ and $\triangle BCD$ have side $CD$ in common, and side $AC$ of $\triangle ACD$ equals side $BC$ of $\triangle BCD$.
The angle $\angle ACD$ ... | Bisection of Straight Line | https://proofwiki.org/wiki/Bisection_of_Straight_Line | https://proofwiki.org/wiki/Bisection_of_Straight_Line | [
"Lines"
] | [
"Definition:Bisection",
"Definition:Line/Segment"
] | [
"Definition:Triangle (Geometry)/Equilateral",
"Definition:Triangle (Geometry)",
"Definition:Subtend",
"Definition:Subtend",
"Bisection of Angle",
"Triangle Side-Angle-Side Congruence",
"Definition:Bisection"
] |
proofwiki-1560 | External Angle of Triangle is Greater than Internal Opposite | The external angle of a triangle is greater than either of the opposite internal angles.
{{:Euclid:Proposition/I/16}} | :250px
Let $\triangle ABC$ be a triangle.
Let the side $BC$ be extended to $D$.
Let $AC$ be bisected at $E$.
Let $BE$ be joined and extended to $F$.
Let $EF$ be made equal to $BE$.
(Technically we really need to extend $BE$ to a point beyond $F$ and then crimp off a length $EF$.)
Let $CF$ be joined.
Let $AC$ be extende... | The [[Definition:External Angle|external angle]] of a [[Definition:Triangle (Geometry)|triangle]] is greater than either of the [[Definition:Opposite (in Triangle)|opposite]] [[Definition:Internal Angle|internal angles]].
{{:Euclid:Proposition/I/16}} | :[[File:Euclid-I-16.png|250px]]
Let $\triangle ABC$ be a [[Definition:Triangle (Geometry)|triangle]].
Let the side $BC$ be [[Axiom:Euclid's Second Postulate|extended to $D$]].
Let $AC$ be [[Bisection of Straight Line|bisected]] at $E$.
Let $BE$ be [[Axiom:Euclid's First Postulate|joined]] and [[Axiom:Euclid's Secon... | External Angle of Triangle is Greater than Internal Opposite | https://proofwiki.org/wiki/External_Angle_of_Triangle_is_Greater_than_Internal_Opposite | https://proofwiki.org/wiki/External_Angle_of_Triangle_is_Greater_than_Internal_Opposite | [
"Triangles",
"External Angles"
] | [
"Definition:Polygon/External Angle",
"Definition:Triangle (Geometry)",
"Definition:Triangle (Geometry)/Opposite",
"Definition:Polygon/Internal Angle"
] | [
"File:Euclid-I-16.png",
"Definition:Triangle (Geometry)",
"Axiom:Euclid's Second Postulate",
"Bisection of Straight Line",
"Axiom:Euclid's First Postulate",
"Axiom:Euclid's Second Postulate",
"Construction of Equal Straight Lines from Unequal",
"Axiom:Euclid's First Postulate",
"Axiom:Euclid's Secon... |
proofwiki-1561 | Limit iff Limits from Left and Right | Let $f$ be a real function defined on an open interval $\openint a b$ except possibly at a point $c \in \openint a b$.
Then:
:$\map f x \to l$ as $x \to c$
{{iff}}:
:$\map f x \to l$ as $x \to c^-$
and
:$\map f x \to l$ as $x \to c^+$ | === Necessary Condition ===
Let $\map f x \to l$ as $x \to c$.
Then from the definition of the limit of a function:
:$\forall \epsilon > 0: \exists \delta > 0: 0 < \size {x - c} < \delta \implies \size {\map f x - l} < \epsilon$
So for any given $\epsilon$, there exists a $\delta$ such that:
:$0 < \size {x - c} < \delt... | Let $f$ be a [[Definition:Real Function|real function]] defined on an [[Definition:Open Real Interval|open interval]] $\openint a b$ except possibly at a point $c \in \openint a b$.
Then:
:$\map f x \to l$ as $x \to c$
{{iff}}:
:$\map f x \to l$ as $x \to c^-$
and
:$\map f x \to l$ as $x \to c^+$ | === Necessary Condition ===
Let $\map f x \to l$ as $x \to c$.
Then from the definition of the [[Definition:Limit of Real Function|limit of a function]]:
:$\forall \epsilon > 0: \exists \delta > 0: 0 < \size {x - c} < \delta \implies \size {\map f x - l} < \epsilon$
So for any given $\epsilon$, there exists a $\delt... | Limit iff Limits from Left and Right | https://proofwiki.org/wiki/Limit_iff_Limits_from_Left_and_Right | https://proofwiki.org/wiki/Limit_iff_Limits_from_Left_and_Right | [
"Limits of Real Functions"
] | [
"Definition:Real Function",
"Definition:Real Interval/Open"
] | [
"Definition:Limit of Real Function",
"Definition:Limit of Real Function/Left",
"Definition:Limit of Real Function/Right"
] |
proofwiki-1562 | Limit of Function by Convergent Sequences | Let $M_1 = \struct {A_1, d_1}$ and $M_2 = \struct {A_2, d_2}$ be metric spaces.
Let $S \subseteq A_1$ be an open set of $M_1$.
Let $f: S \to A_2$ be a mapping defined on $S$, except possibly at the point $c \in S$.
Then $\ds \lim_{x \mathop \to c} \map f x = l$ {{iff}}:
:for each sequence $\sequence {x_n}$ of points of... | === Necessary Condition ===
Suppose that:
:$\ds \lim_{x \mathop \to c} \map f x = l$
Let $\epsilon > 0$.
Then by the definition of the limit of a mapping:
:$\exists \delta > 0: \map {d_2} {\map f x, l} < \epsilon$
provided $0 < \map {d_1} {x, c} < \delta$.
Now suppose that $\sequence {x_n}$ is a sequence of points of $... | Let $M_1 = \struct {A_1, d_1}$ and $M_2 = \struct {A_2, d_2}$ be [[Definition:Metric Space|metric spaces]].
Let $S \subseteq A_1$ be an [[Definition:Open Set (Metric Space)|open set]] of $M_1$.
Let $f: S \to A_2$ be a [[Definition:Mapping|mapping]] defined on $S$, except possibly at the point $c \in S$.
Then $\ds \... | === Necessary Condition ===
Suppose that:
:$\ds \lim_{x \mathop \to c} \map f x = l$
Let $\epsilon > 0$.
Then by the definition of the [[Definition:Limit of Mapping between Metric Spaces|limit of a mapping]]:
:$\exists \delta > 0: \map {d_2} {\map f x, l} < \epsilon$
provided $0 < \map {d_1} {x, c} < \delta$.
Now s... | Limit of Function by Convergent Sequences | https://proofwiki.org/wiki/Limit_of_Function_by_Convergent_Sequences | https://proofwiki.org/wiki/Limit_of_Function_by_Convergent_Sequences | [
"Metric Spaces",
"Limits of Sequences"
] | [
"Definition:Metric Space",
"Definition:Open Set/Metric Space",
"Definition:Mapping",
"Definition:Sequence"
] | [
"Definition:Limit of Mapping between Metric Spaces",
"Definition:Sequence",
"Definition:Limit of Sequence/Metric Space",
"Definition:Metric Space/Metric",
"Definition:Sequence",
"Definition:Sequence"
] |
proofwiki-1563 | Real Polynomial Function is Continuous | A (real) polynomial function is continuous at every point.
Thus a (real) polynomial function is continuous on every interval of $\R$. | Let $f_n$ be an arbitrary real polynomial function of degree $n$.
The proof proceeds by induction.
For all $n \in \Z_{\ge 0}$, let $\map P n$ be the proposition:
:$f_n$ is continuous on $\R$.
$\map P 0$ is the case $f_0$, where $f_0$ is of zero degree.
Such a real polynomial function is a constant function.
It follows ... | A [[Definition:Real Polynomial Function|(real) polynomial function]] is [[Definition:Continuous Real Function at Point|continuous]] at every [[Definition:Point|point]].
Thus a [[Definition:Real Polynomial Function|(real) polynomial function]] is [[Definition:Continuous Real Function on Interval|continuous]] on every [... | Let $f_n$ be an arbitrary [[Definition:Real Polynomial Function|real polynomial function]] of [[Definition:Degree of Polynomial|degree]] $n$.
The proof proceeds by [[Principle of Mathematical Induction|induction]].
For all $n \in \Z_{\ge 0}$, let $\map P n$ be the [[Definition:Proposition|proposition]]:
:$f_n$ is [[D... | Real Polynomial Function is Continuous/Proof 1 | https://proofwiki.org/wiki/Real_Polynomial_Function_is_Continuous | https://proofwiki.org/wiki/Real_Polynomial_Function_is_Continuous/Proof_1 | [
"Real Polynomial Function is Continuous",
"Continuous Real Functions",
"Real Polynomial Functions"
] | [
"Definition:Polynomial Function/Real",
"Definition:Continuous Real Function/Point",
"Definition:Point",
"Definition:Polynomial Function/Real",
"Definition:Continuous Real Function/Interval",
"Definition:Real Interval"
] | [
"Definition:Polynomial Function/Real",
"Definition:Degree of Polynomial",
"Principle of Mathematical Induction",
"Definition:Proposition",
"Definition:Continuous Real Function",
"Definition:Degree of Polynomial/Zero",
"Definition:Polynomial Function/Real",
"Definition:Constant Mapping",
"Constant Fu... |
proofwiki-1564 | Linear Function is Continuous | Let $\alpha, \beta \in \R$ be real numbers.
Let $f : \R \to \R$ be a linear real function:
:$\map f x = \alpha x + \beta$
for all $x \in \R$.
Then $f$ is continuous at every real number $c \in \R$. | First assume $\alpha \ne 0$.
Let $\epsilon \in \R_{>0}$.
Let $\delta = \dfrac \epsilon {\size \alpha}$.
Then, provided that $\size {x - c} < \delta$:
{{begin-eqn}}
{{eqn | l = \size {\map f x - \map f c}
| r = \size {\alpha \paren {x - c} }
| c =
}}
{{eqn | r = \size {\alpha} \cdot \size {x - c}
| c ... | Let $\alpha, \beta \in \R$ be [[Definition:Real Number|real numbers]].
Let $f : \R \to \R$ be a [[Definition:Linear Real Function|linear real function]]:
:$\map f x = \alpha x + \beta$
for all $x \in \R$.
Then $f$ is [[Definition:Continuous Real Function at Point|continuous]] at every real number $c \in \R$. | First assume $\alpha \ne 0$.
Let $\epsilon \in \R_{>0}$.
Let $\delta = \dfrac \epsilon {\size \alpha}$.
Then, provided that $\size {x - c} < \delta$:
{{begin-eqn}}
{{eqn | l = \size {\map f x - \map f c}
| r = \size {\alpha \paren {x - c} }
| c =
}}
{{eqn | r = \size {\alpha} \cdot \size {x - c}
... | Linear Function is Continuous/Proof 1 | https://proofwiki.org/wiki/Linear_Function_is_Continuous | https://proofwiki.org/wiki/Linear_Function_is_Continuous/Proof_1 | [
"Linear Real Functions",
"Continuous Real Functions",
"Linear Function is Continuous"
] | [
"Definition:Real Number",
"Definition:Linear Real Function",
"Definition:Continuous Real Function/Point"
] | [
"Definition:Continuous Real Function/Point"
] |
proofwiki-1565 | Linear Function is Continuous | Let $\alpha, \beta \in \R$ be real numbers.
Let $f : \R \to \R$ be a linear real function:
:$\map f x = \alpha x + \beta$
for all $x \in \R$.
Then $f$ is continuous at every real number $c \in \R$. | Let $c \in \R$.
Let $\sequence {x_n}$ be a real sequence converging to $c$.
Then:
{{begin-eqn}}
{{eqn | l = \lim_{n \mathop \to \infty} \map f {x_n}
| r = \lim_{n \mathop \to \infty} \paren {\alpha x_n + \beta}
}}
{{eqn | r = \alpha c + \beta
| c = Combined Sum Rule for Real Sequences
}}
{{eqn | r = \map f c
}}
{{e... | Let $\alpha, \beta \in \R$ be [[Definition:Real Number|real numbers]].
Let $f : \R \to \R$ be a [[Definition:Linear Real Function|linear real function]]:
:$\map f x = \alpha x + \beta$
for all $x \in \R$.
Then $f$ is [[Definition:Continuous Real Function at Point|continuous]] at every real number $c \in \R$. | Let $c \in \R$.
Let $\sequence {x_n}$ be a [[Definition:Real Sequence|real sequence]] [[Definition:Convergent Real Sequence|converging]] to $c$.
Then:
{{begin-eqn}}
{{eqn | l = \lim_{n \mathop \to \infty} \map f {x_n}
| r = \lim_{n \mathop \to \infty} \paren {\alpha x_n + \beta}
}}
{{eqn | r = \alpha c + \beta
|... | Linear Function is Continuous/Proof 2 | https://proofwiki.org/wiki/Linear_Function_is_Continuous | https://proofwiki.org/wiki/Linear_Function_is_Continuous/Proof_2 | [
"Linear Real Functions",
"Continuous Real Functions",
"Linear Function is Continuous"
] | [
"Definition:Real Number",
"Definition:Linear Real Function",
"Definition:Continuous Real Function/Point"
] | [
"Definition:Real Sequence",
"Definition:Convergent Sequence/Real Numbers",
"Combination Theorem for Sequences/Real/Combined Sum Rule",
"Definition:Real Sequence",
"Definition:Convergent Sequence/Real Numbers",
"Sequential Continuity is Equivalent to Continuity in the Reals",
"Definition:Continuous Real ... |
proofwiki-1566 | Real Rational Function is Continuous | A real rational function is continuous at every point at which it is defined.
Thus a real rational function is continuous on every interval of $\R$ not containing a root of the denominator of the function. | Let:
:$\map R x = \dfrac {\map P x} {\map Q x}$
be a real rational function, defined at all points of $\R$ at which $\map Q x \ne 0$.
Let $c \in \R$.
From Real Polynomial Function is Continuous:
:$\ds \lim_{x \mathop \to c} \map P x = \map P c$
and:
:$\ds \lim_{x \mathop \to c} \map Q x = \map Q c$
Thus by Quotient Rul... | A [[Definition:Real Function|real]] [[Definition:Rational Function|rational function]] is [[Definition:Continuous Real Function|continuous]] at every point at which it is defined.
Thus a [[Definition:Real Function|real]] [[Definition:Rational Function|rational function]] is [[Definition:Continuous on Interval|continuo... | Let:
:$\map R x = \dfrac {\map P x} {\map Q x}$
be a [[Definition:Real Function|real]] [[Definition:Rational Function|rational function]], defined at all points of $\R$ at which $\map Q x \ne 0$.
Let $c \in \R$.
From [[Real Polynomial Function is Continuous]]:
:$\ds \lim_{x \mathop \to c} \map P x = \map P c$
and:
:$... | Real Rational Function is Continuous | https://proofwiki.org/wiki/Real_Rational_Function_is_Continuous | https://proofwiki.org/wiki/Real_Rational_Function_is_Continuous | [
"Real Analysis",
"Continuous Real Functions"
] | [
"Definition:Real Function",
"Definition:Rational Function",
"Definition:Continuous Real Function",
"Definition:Real Function",
"Definition:Rational Function",
"Definition:Continuous Real Function/Interval",
"Definition:Real Interval",
"Definition:Root of Polynomial",
"Definition:Fraction/Denominator... | [
"Definition:Real Function",
"Definition:Rational Function",
"Real Polynomial Function is Continuous",
"Combination Theorem for Limits of Functions/Real/Quotient Rule",
"Definition:Continuous Real Function",
"Definition:Continuous Real Function/Interval"
] |
proofwiki-1567 | Limit of Composite Function | Let $f$ and $g$ be real functions.
Let:
{{begin-eqn}}
{{eqn | l = \ds \lim_{y \mathop \to \eta} \map f y
| r = l
}}
{{eqn | l = \ds \lim_{x \mathop \to \xi} \map g x
| r = \eta
}}
{{end-eqn}}
Then, if either:
:'''Hypothesis 1:''' $f$ is continuous at $\eta$ (that is $l = \map f \eta$)
or:
:'''Hypothesis 2:'... | Let $\epsilon > 0$.
Since $\ds \lim_{y \mathop \to \eta} \map f y = l$, we can find $\Delta > 0$ such that:
:$\size {\map f y - l} < \epsilon$ provided $0 < \size {y - \eta} < \Delta$
Let $y = \map g x$.
Then, provided that $0 < \size {\map g x - \eta} < \Delta$, we have:
:$\size {\map f {\map g x} - l} < \epsilon$
But... | Let $f$ and $g$ be [[Definition:Real Function|real functions]].
Let:
{{begin-eqn}}
{{eqn | l = \ds \lim_{y \mathop \to \eta} \map f y
| r = l
}}
{{eqn | l = \ds \lim_{x \mathop \to \xi} \map g x
| r = \eta
}}
{{end-eqn}}
Then, if either:
:'''Hypothesis 1:''' $f$ is [[Definition:Continuous Real Function ... | Let $\epsilon > 0$.
Since $\ds \lim_{y \mathop \to \eta} \map f y = l$, we can find $\Delta > 0$ such that:
:$\size {\map f y - l} < \epsilon$ provided $0 < \size {y - \eta} < \Delta$
Let $y = \map g x$.
Then, provided that $0 < \size {\map g x - \eta} < \Delta$, we have:
:$\size {\map f {\map g x} - l} < \epsilon$
... | Limit of Composite Function | https://proofwiki.org/wiki/Limit_of_Composite_Function | https://proofwiki.org/wiki/Limit_of_Composite_Function | [
"Limit of Composite Function",
"Limits of Real Functions"
] | [
"Definition:Real Function",
"Definition:Continuous Real Function/Point",
"Definition:Real Interval/Open"
] | [] |
proofwiki-1568 | Condition for Continuity on Interval | Let $f$ be a real function defined on an interval $\mathbb I$.
Then $f$ is continuous on $\mathbb I$ {{iff}}:
:$\forall x \in \mathbb I: \forall \epsilon > 0: \exists \delta > 0: y \in \mathbb I \land \size {x - y} < \delta \implies \size {\map f x - \map f y} < \epsilon$ | Let $x \in \mathbb I$ such that $x$ is not an end point.
Then the condition $y \in \mathbb I \land \size {x - y} < \delta$ is the same as $\size {x - y} < \delta$ provided $\delta$ is small enough.
The criterion given therefore becomes the same as the statement $\ds \lim_{y \mathop \to x} \map f y = \map f x$, that is,... | Let $f$ be a [[Definition:Real Function|real function]] defined on an [[Definition:Real Interval|interval]] $\mathbb I$.
Then $f$ is [[Definition:Continuous on Interval|continuous]] on $\mathbb I$ {{iff}}:
:$\forall x \in \mathbb I: \forall \epsilon > 0: \exists \delta > 0: y \in \mathbb I \land \size {x - y} < \delt... | Let $x \in \mathbb I$ such that $x$ is not an [[Definition:Endpoint of Real Interval|end point]].
Then the condition $y \in \mathbb I \land \size {x - y} < \delta$ is the same as $\size {x - y} < \delta$ provided $\delta$ is small enough.
The criterion given therefore becomes the same as the statement $\ds \lim_{y \m... | Condition for Continuity on Interval | https://proofwiki.org/wiki/Condition_for_Continuity_on_Interval | https://proofwiki.org/wiki/Condition_for_Continuity_on_Interval | [
"Real Intervals",
"Continuous Real Functions"
] | [
"Definition:Real Function",
"Definition:Real Interval",
"Definition:Continuous Real Function/Interval"
] | [
"Definition:Real Interval/Endpoints",
"Definition:Continuous Real Function/Point",
"Definition:Real Interval/Endpoints",
"Definition:Continuous Real Function/Right-Continuous",
"Definition:Real Interval/Endpoints",
"Definition:Continuous Real Function/Left-Continuous",
"Definition:Continuous Real Functi... |
proofwiki-1569 | Limit of Image of Sequence | Let $M_1 = \struct {A_1, d_1}$ and $M_2 = \struct {A_2, d_2}$ be metric spaces.
Let $f: A_1 \to A_2$ be a mapping which is continuous at $a \in A_1$.
Let $\sequence {x_n}$ be a sequence of points in $A_1$ such that:
:$\ds \lim_{n \mathop \to \infty} x_n = a$
where $\ds \lim_{n \mathop \to \infty} x_n$ is the limit of $... | From Limit of Function by Convergent Sequences, we have:
:$\ds \lim_{x \mathop \to a} \map f x = \map f a$
{{iff}}:
:for each sequence $\sequence {x_n}$ of points of $A_1$ such that:
::$\forall n \in \N_{>0}: x_n \ne a$
:and:
::$\ds \lim_{n \mathop \to \infty} x_n = a$
:it is true that:
::$\ds \lim_{n \mathop \to \inft... | Let $M_1 = \struct {A_1, d_1}$ and $M_2 = \struct {A_2, d_2}$ be [[Definition:Metric Space|metric spaces]].
Let $f: A_1 \to A_2$ be a [[Definition:Mapping|mapping]] which is [[Definition:Continuous Mapping (Metric Spaces)|continuous]] at $a \in A_1$.
Let $\sequence {x_n}$ be a [[Definition:Sequence|sequence]] of poin... | From [[Limit of Function by Convergent Sequences]], we have:
:$\ds \lim_{x \mathop \to a} \map f x = \map f a$
{{iff}}:
:for each [[Definition:Sequence|sequence]] $\sequence {x_n}$ of points of $A_1$ such that:
::$\forall n \in \N_{>0}: x_n \ne a$
:and:
::$\ds \lim_{n \mathop \to \infty} x_n = a$
:it is true that:
::$... | Limit of Image of Sequence | https://proofwiki.org/wiki/Limit_of_Image_of_Sequence | https://proofwiki.org/wiki/Limit_of_Image_of_Sequence | [
"Continuous Mappings on Metric Spaces",
"Limits of Sequences"
] | [
"Definition:Metric Space",
"Definition:Mapping",
"Definition:Continuous Mapping (Metric Space)",
"Definition:Sequence",
"Definition:Limit of Sequence/Metric Space",
"Definition:Continuous Mapping (Metric Space)",
"Definition:Commutative/Elements"
] | [
"Limit of Function by Convergent Sequences",
"Definition:Sequence",
"Definition:Continuous Mapping (Metric Space)"
] |
proofwiki-1570 | Image of Real Interval under Continuous Real Function is Real Interval | Let $I$ be a real interval.
Let $f: I \to \R$ be a continuous real function.
Then the image of $f$ is a real interval. | Let $J$ be the image of $f$.
By definition of real interval, it suffices to show that:
:$\forall y_1, y_2 \in J: \forall \lambda \in \R: y_1 \le \lambda \le y_2 \implies \lambda \in J$
So suppose $y_1, y_2 \in J$, and suppose $\lambda \in \R$ is such that $y_1 \le \lambda \le y_2$.
Consider these subsets of $I$:
:$S = ... | Let $I$ be a [[Definition:Real Interval|real interval]].
Let $f: I \to \R$ be a [[Definition:Continuous on Interval|continuous]] [[Definition:Real Function|real function]].
Then the [[Definition:Image of Mapping|image]] of $f$ is a [[Definition:Real Interval|real interval]]. | Let $J$ be the [[Definition:Image of Mapping|image]] of $f$.
By definition of [[Definition:Real Interval|real interval]], it suffices to show that:
:$\forall y_1, y_2 \in J: \forall \lambda \in \R: y_1 \le \lambda \le y_2 \implies \lambda \in J$
So suppose $y_1, y_2 \in J$, and suppose $\lambda \in \R$ is such that ... | Image of Real Interval under Continuous Real Function is Real Interval/Proof 1 | https://proofwiki.org/wiki/Image_of_Real_Interval_under_Continuous_Real_Function_is_Real_Interval | https://proofwiki.org/wiki/Image_of_Real_Interval_under_Continuous_Real_Function_is_Real_Interval/Proof_1 | [
"Image of Real Interval under Continuous Real Function is Real Interval",
"Continuous Real Functions",
"Real Intervals"
] | [
"Definition:Real Interval",
"Definition:Continuous Real Function/Interval",
"Definition:Real Function",
"Definition:Image (Set Theory)/Mapping/Mapping",
"Definition:Real Interval"
] | [
"Definition:Image (Set Theory)/Mapping/Mapping",
"Definition:Real Interval",
"Definition:Non-Empty Set",
"Interval Divided into Subsets",
"Distance from Subset of Real Numbers",
"Distance from Subset of Real Numbers",
"Limit of Sequence to Zero Distance Point",
"Definition:Sequence",
"Definition:Con... |
proofwiki-1571 | Image of Real Interval under Continuous Real Function is Real Interval | Let $I$ be a real interval.
Let $f: I \to \R$ be a continuous real function.
Then the image of $f$ is a real interval. | Let $J$ be the image of $f$.
By Subset of Real Numbers is Interval iff Connected we need to show that $J$ is connected (and hence an interval).
{{AimForCont}} not.
Then there exists a separation $S \mid T$ of $J$.
Define $A = f^{-1} \sqbrk S$ and $B = f^{-1} \sqbrk T$. $A$ and $B$ are both non-empty.
Because $f$ is con... | Let $I$ be a [[Definition:Real Interval|real interval]].
Let $f: I \to \R$ be a [[Definition:Continuous on Interval|continuous]] [[Definition:Real Function|real function]].
Then the [[Definition:Image of Mapping|image]] of $f$ is a [[Definition:Real Interval|real interval]]. | Let $J$ be the [[Definition:Image of Mapping|image]] of $f$.
By [[Subset of Real Numbers is Interval iff Connected]] we need to show that $J$ is connected (and hence an interval).
{{AimForCont}} not.
Then there exists a [[Definition:Separation (Topology)|separation]] $S \mid T$ of $J$.
Define $A = f^{-1} \sqbrk S$ ... | Image of Real Interval under Continuous Real Function is Real Interval/Proof 2 | https://proofwiki.org/wiki/Image_of_Real_Interval_under_Continuous_Real_Function_is_Real_Interval | https://proofwiki.org/wiki/Image_of_Real_Interval_under_Continuous_Real_Function_is_Real_Interval/Proof_2 | [
"Image of Real Interval under Continuous Real Function is Real Interval",
"Continuous Real Functions",
"Real Intervals"
] | [
"Definition:Real Interval",
"Definition:Continuous Real Function/Interval",
"Definition:Real Function",
"Definition:Image (Set Theory)/Mapping/Mapping",
"Definition:Real Interval"
] | [
"Definition:Image (Set Theory)/Mapping/Mapping",
"Subset of Real Numbers is Interval iff Connected",
"Definition:Separation (Topology)",
"Continuous iff inverse image of any open set is open"
] |
proofwiki-1572 | Interval Divided into Subsets | Let $\mathbb I$ be a real interval.
Let $S$ and $T$ be non-empty subsets of $\mathbb I$ such that $\mathbb I \subseteq S \cup T$.
Then one of $S$ or $T$ contains an element at zero distance from the other. | $\mathbb I \subseteq S \cup T \implies \forall x \in \mathbb I: x \in S \lor x \in T$ from the definition of union.
That is, every element of $\mathbb I$ belongs either to $S$ or to $T$.
The distance of an element $c \in \R$ from a subset $S$ of $\R$ is given as:
:$\ds \map d {c, S} = \map {\inf_{x \mathop \in S} } {\s... | Let $\mathbb I$ be a [[Definition:Real Interval|real interval]].
Let $S$ and $T$ be [[Definition:Non-Empty Set|non-empty]] subsets of $\mathbb I$ such that $\mathbb I \subseteq S \cup T$.
Then one of $S$ or $T$ contains an [[Definition:Element|element]] at zero [[Distance from Subset of Real Numbers|distance]] from ... | $\mathbb I \subseteq S \cup T \implies \forall x \in \mathbb I: x \in S \lor x \in T$ from the definition of [[Definition:Set Union|union]].
That is, every [[Definition:Element|element]] of $\mathbb I$ belongs either to $S$ or to $T$.
The [[Distance from Subset of Real Numbers|distance]] of an [[Definition:Element|e... | Interval Divided into Subsets | https://proofwiki.org/wiki/Interval_Divided_into_Subsets | https://proofwiki.org/wiki/Interval_Divided_into_Subsets | [
"Real Analysis"
] | [
"Definition:Real Interval",
"Definition:Non-Empty Set",
"Definition:Element",
"Distance from Subset of Real Numbers"
] | [
"Definition:Set Union",
"Definition:Element",
"Distance from Subset of Real Numbers",
"Definition:Element",
"Definition:Subset",
"Definition:Element",
"Definition:Bounded Below Set",
"Distance from Subset of Real Numbers",
"Definition:Element",
"Distance from Subset of Real Numbers",
"Definition... |
proofwiki-1573 | Distance from Subset of Real Numbers | Let $S$ be a subset of the set of real numbers $\R$.
Let $x \in \R$ be a real number.
Let $\map d {x, S}$ be the distance between $x$ and $S$.
Then: | From the definition of distance:
:$\forall x, y \in \R: \map d {x, y} = \size {x - y}$
Thus:
:$\ds \map d {x, S} = \map {\inf_{y \mathop \in S} } {\size {x - y} }$
Consider the set $T = \set {\size {x - y}: y \in S}$.
This has $0$ as a lower bound as Absolute Value is Bounded Below by Zero.
So:
:$\ds \map d {x, S} = \m... | Let $S$ be a [[Definition:Subset|subset]] of the set of [[Definition:Real Number|real numbers]] $\R$.
Let $x \in \R$ be a [[Definition:Real Number|real number]].
Let $\map d {x, S}$ be the [[Definition:Distance between Element and Subset of Real Numbers|distance]] between $x$ and $S$.
Then: | From the definition of [[Definition:Distance between Real Numbers|distance]]:
:$\forall x, y \in \R: \map d {x, y} = \size {x - y}$
Thus:
:$\ds \map d {x, S} = \map {\inf_{y \mathop \in S} } {\size {x - y} }$
Consider the set $T = \set {\size {x - y}: y \in S}$.
This has $0$ as a [[Definition:Lower Bound of Set|low... | Distance from Subset of Real Numbers to Element/Proof 1 | https://proofwiki.org/wiki/Distance_from_Subset_of_Real_Numbers | https://proofwiki.org/wiki/Distance_from_Subset_of_Real_Numbers_to_Element/Proof_1 | [
"Distance Function"
] | [
"Definition:Subset",
"Definition:Real Number",
"Definition:Real Number",
"Definition:Distance/Sets/Real Numbers"
] | [
"Definition:Distance/Points/Real Numbers",
"Definition:Lower Bound of Set",
"Absolute Value is Bounded Below by Zero"
] |
proofwiki-1574 | Distance from Subset of Real Numbers | Let $S$ be a subset of the set of real numbers $\R$.
Let $x \in \R$ be a real number.
Let $\map d {x, S}$ be the distance between $x$ and $S$.
Then: | Recall from Real Number Line is Metric Space that the set of real numbers $\R$ with the distance function $d$ is a metric space.
The result is then seen to be an example of Distance from Subset to Element.
{{Qed}} | Let $S$ be a [[Definition:Subset|subset]] of the set of [[Definition:Real Number|real numbers]] $\R$.
Let $x \in \R$ be a [[Definition:Real Number|real number]].
Let $\map d {x, S}$ be the [[Definition:Distance between Element and Subset of Real Numbers|distance]] between $x$ and $S$.
Then: | Recall from [[Real Number Line is Metric Space]] that the [[Definition:Real Number|set of real numbers]] $\R$ with the [[Definition:Distance between Element and Subset of Real Numbers|distance function]] $d$ is a [[Definition:Metric Space|metric space]].
The result is then seen to be an example of [[Distance from Subs... | Distance from Subset of Real Numbers to Element/Proof 2 | https://proofwiki.org/wiki/Distance_from_Subset_of_Real_Numbers | https://proofwiki.org/wiki/Distance_from_Subset_of_Real_Numbers_to_Element/Proof_2 | [
"Distance Function"
] | [
"Definition:Subset",
"Definition:Real Number",
"Definition:Real Number",
"Definition:Distance/Sets/Real Numbers"
] | [
"Real Number Line is Metric Space",
"Definition:Real Number",
"Definition:Distance/Sets/Real Numbers",
"Definition:Metric Space",
"Distance from Subset to Element"
] |
proofwiki-1575 | Distance from Subset of Real Numbers | Let $S$ be a subset of the set of real numbers $\R$.
Let $x \in \R$ be a real number.
Let $\map d {x, S}$ be the distance between $x$ and $S$.
Then: | From the definition of distance:
:$\forall x, y \in \R: \map d {x, y} = \size {x - y}$
Thus:
:$\ds \map d {x, S} = \map {\inf_{y \mathop \in S} } {\size {x - y} }$
Let $\xi = \inf S$.
Consider $\map d {-\xi, S'}$ where $S' = \set {-\xi: \xi \in S}$.
By Negative of Infimum is Supremum of Negatives:
:$\xi = \inf S \impli... | Let $S$ be a [[Definition:Subset|subset]] of the set of [[Definition:Real Number|real numbers]] $\R$.
Let $x \in \R$ be a [[Definition:Real Number|real number]].
Let $\map d {x, S}$ be the [[Definition:Distance between Element and Subset of Real Numbers|distance]] between $x$ and $S$.
Then: | From the definition of [[Definition:Distance between Real Numbers|distance]]:
:$\forall x, y \in \R: \map d {x, y} = \size {x - y}$
Thus:
:$\ds \map d {x, S} = \map {\inf_{y \mathop \in S} } {\size {x - y} }$
Let $\xi = \inf S$.
Consider $\map d {-\xi, S'}$ where $S' = \set {-\xi: \xi \in S}$.
By [[Negative of Inf... | Distance from Subset of Real Numbers to Infimum/Proof 1 | https://proofwiki.org/wiki/Distance_from_Subset_of_Real_Numbers | https://proofwiki.org/wiki/Distance_from_Subset_of_Real_Numbers_to_Infimum/Proof_1 | [
"Distance Function"
] | [
"Definition:Subset",
"Definition:Real Number",
"Definition:Real Number",
"Definition:Distance/Sets/Real Numbers"
] | [
"Definition:Distance/Points/Real Numbers",
"Negative of Infimum is Supremum of Negatives"
] |
proofwiki-1576 | Distance from Subset of Real Numbers | Let $S$ be a subset of the set of real numbers $\R$.
Let $x \in \R$ be a real number.
Let $\map d {x, S}$ be the distance between $x$ and $S$.
Then: | Recall from Real Number Line is Metric Space that the set of real numbers $\R$ with the distance function $d$ is a metric space.
The result is then seen to be an example of Distance from Subset to Infimum.
{{Qed}} | Let $S$ be a [[Definition:Subset|subset]] of the set of [[Definition:Real Number|real numbers]] $\R$.
Let $x \in \R$ be a [[Definition:Real Number|real number]].
Let $\map d {x, S}$ be the [[Definition:Distance between Element and Subset of Real Numbers|distance]] between $x$ and $S$.
Then: | Recall from [[Real Number Line is Metric Space]] that the [[Definition:Real Number|set of real numbers]] $\R$ with the [[Definition:Distance between Element and Subset of Real Numbers|distance function]] $d$ is a [[Definition:Metric Space|metric space]].
The result is then seen to be an example of [[Distance from Subs... | Distance from Subset of Real Numbers to Infimum/Proof 2 | https://proofwiki.org/wiki/Distance_from_Subset_of_Real_Numbers | https://proofwiki.org/wiki/Distance_from_Subset_of_Real_Numbers_to_Infimum/Proof_2 | [
"Distance Function"
] | [
"Definition:Subset",
"Definition:Real Number",
"Definition:Real Number",
"Definition:Distance/Sets/Real Numbers"
] | [
"Real Number Line is Metric Space",
"Definition:Real Number",
"Definition:Distance/Sets/Real Numbers",
"Definition:Metric Space",
"Distance from Subset to Infimum"
] |
proofwiki-1577 | Distance from Subset of Real Numbers | Let $S$ be a subset of the set of real numbers $\R$.
Let $x \in \R$ be a real number.
Let $\map d {x, S}$ be the distance between $x$ and $S$.
Then: | From the definition of distance:
:$\forall x, y \in \R: \map d {x, y} = \size {x - y}$
Thus:
:$\ds \map d {x, S} = \map {\inf_{y \mathop \in S} } {\size {x - y} }$
Let $\xi = \sup S$.
Then:
:$\forall y \in S: \size {\xi - y} = \xi - y$
So we need to show that no $h > 0$ can be a lower bound for $T = \set {\size {\xi - ... | Let $S$ be a [[Definition:Subset|subset]] of the set of [[Definition:Real Number|real numbers]] $\R$.
Let $x \in \R$ be a [[Definition:Real Number|real number]].
Let $\map d {x, S}$ be the [[Definition:Distance between Element and Subset of Real Numbers|distance]] between $x$ and $S$.
Then: | From the definition of [[Definition:Distance between Real Numbers|distance]]:
:$\forall x, y \in \R: \map d {x, y} = \size {x - y}$
Thus:
:$\ds \map d {x, S} = \map {\inf_{y \mathop \in S} } {\size {x - y} }$
Let $\xi = \sup S$.
Then:
:$\forall y \in S: \size {\xi - y} = \xi - y$
So we need to show that no $h > 0$... | Distance from Subset of Real Numbers to Supremum/Proof 1 | https://proofwiki.org/wiki/Distance_from_Subset_of_Real_Numbers | https://proofwiki.org/wiki/Distance_from_Subset_of_Real_Numbers_to_Supremum/Proof_1 | [
"Distance Function"
] | [
"Definition:Subset",
"Definition:Real Number",
"Definition:Real Number",
"Definition:Distance/Sets/Real Numbers"
] | [
"Definition:Distance/Points/Real Numbers",
"Definition:Lower Bound of Set",
"Definition:Upper Bound of Set",
"Definition:Contradiction",
"Definition:Supremum of Set",
"Definition:Upper Bound of Set",
"Proof by Contradiction"
] |
proofwiki-1578 | Distance from Subset of Real Numbers | Let $S$ be a subset of the set of real numbers $\R$.
Let $x \in \R$ be a real number.
Let $\map d {x, S}$ be the distance between $x$ and $S$.
Then: | Recall from Real Number Line is Metric Space that the set of real numbers $\R$ with the distance function $d$ is a metric space.
The result is then seen to be an example of Distance from Subset to Supremum.
{{Qed}} | Let $S$ be a [[Definition:Subset|subset]] of the set of [[Definition:Real Number|real numbers]] $\R$.
Let $x \in \R$ be a [[Definition:Real Number|real number]].
Let $\map d {x, S}$ be the [[Definition:Distance between Element and Subset of Real Numbers|distance]] between $x$ and $S$.
Then: | Recall from [[Real Number Line is Metric Space]] that the [[Definition:Real Number|set of real numbers]] $\R$ with the [[Definition:Distance between Element and Subset of Real Numbers|distance function]] $d$ is a [[Definition:Metric Space|metric space]].
The result is then seen to be an example of [[Distance from Subs... | Distance from Subset of Real Numbers to Supremum/Proof 2 | https://proofwiki.org/wiki/Distance_from_Subset_of_Real_Numbers | https://proofwiki.org/wiki/Distance_from_Subset_of_Real_Numbers_to_Supremum/Proof_2 | [
"Distance Function"
] | [
"Definition:Subset",
"Definition:Real Number",
"Definition:Real Number",
"Definition:Distance/Sets/Real Numbers"
] | [
"Real Number Line is Metric Space",
"Definition:Real Number",
"Definition:Distance/Sets/Real Numbers",
"Definition:Metric Space",
"Distance from Subset to Supremum"
] |
proofwiki-1579 | Limit of Sequence to Zero Distance Point | Let $S$ be a non-empty subset of $\R$.
Let the distance $\map d {\xi, S} = 0$ for some $\xi \in \R$.
Then there exists a sequence $\sequence {x_n}$ in $S$ such that $\ds \lim_{n \mathop \to \infty} x_n = \xi$. | First it is shown that:
:$\forall n \in \N_{>0}: \exists x_n \in S: \size {\xi - x_n} < \dfrac 1 n$
{{AimForCont}} that:
:$\exists n \in \N_{>0}: \not \exists x \in S: \size {\xi - x} < \dfrac 1 n$
Then $\dfrac 1 n$ is a lower bound of the set $T = \set {\size {\xi - x}: x \in S}$.
This contradicts the assertion that $... | Let $S$ be a [[Definition:Non-Empty Set|non-empty]] [[Definition:Subset|subset]] of $\R$.
Let the [[Distance from Subset of Real Numbers|distance]] $\map d {\xi, S} = 0$ for some $\xi \in \R$.
Then there exists a [[Definition:Sequence|sequence]] $\sequence {x_n}$ in $S$ such that $\ds \lim_{n \mathop \to \infty} x_n... | First it is shown that:
:$\forall n \in \N_{>0}: \exists x_n \in S: \size {\xi - x_n} < \dfrac 1 n$
{{AimForCont}} that:
:$\exists n \in \N_{>0}: \not \exists x \in S: \size {\xi - x} < \dfrac 1 n$
Then $\dfrac 1 n$ is a [[Definition:Lower Bound of Subset of Real Numbers|lower bound]] of the set $T = \set {\size {\xi... | Limit of Sequence to Zero Distance Point | https://proofwiki.org/wiki/Limit_of_Sequence_to_Zero_Distance_Point | https://proofwiki.org/wiki/Limit_of_Sequence_to_Zero_Distance_Point | [
"Limits of Sequences",
"Limit of Sequence to Zero Distance Point"
] | [
"Definition:Non-Empty Set",
"Definition:Subset",
"Distance from Subset of Real Numbers",
"Definition:Sequence"
] | [
"Definition:Lower Bound of Set/Real Numbers",
"Proof by Contradiction",
"Sequence of Powers of Reciprocals is Null Sequence",
"Squeeze Theorem/Sequences/Real Numbers"
] |
proofwiki-1580 | Image of Closed Real Interval is Bounded | Let $f$ be a real function which is continuous on the closed interval $\closedint a b$.
Then $f$ is bounded on $\closedint a b$. | {{AimForCont}} $f$ is not bounded on $\closedint a b$.
Then from {{Corollary|Limit of Sequence to Zero Distance Point}}, there exists a sequence $\sequence {x_n}$ in $\closedint a b$ such that $\size {\map f {x_n} } \to +\infty$ as $n \to \infty$.
Since $\closedint a b$ is a closed interval, from Convergent Subsequence... | Let $f$ be a [[Definition:Real Function|real function]] which is [[Definition:Continuous Real Function on Closed Interval|continuous]] on the [[Definition:Closed Real Interval|closed interval]] $\closedint a b$.
Then $f$ is [[Definition:Bounded Mapping|bounded]] on $\closedint a b$. | {{AimForCont}} $f$ is not [[Definition:Bounded Mapping|bounded]] on $\closedint a b$.
Then from {{Corollary|Limit of Sequence to Zero Distance Point}}, there exists a [[Definition:Sequence|sequence]] $\sequence {x_n}$ in $\closedint a b$ such that $\size {\map f {x_n} } \to +\infty$ as $n \to \infty$.
Since $\closedi... | Image of Closed Real Interval is Bounded | https://proofwiki.org/wiki/Image_of_Closed_Real_Interval_is_Bounded | https://proofwiki.org/wiki/Image_of_Closed_Real_Interval_is_Bounded | [
"Analysis"
] | [
"Definition:Real Function",
"Definition:Continuous Real Function/Closed Interval",
"Definition:Real Interval/Closed",
"Definition:Bounded Mapping"
] | [
"Definition:Bounded Mapping",
"Definition:Sequence",
"Definition:Real Interval/Closed",
"Convergent Subsequence in Closed Interval",
"Definition:Subsequence",
"Definition:Convergent Sequence",
"Definition:Continuous Real Function/Closed Interval",
"Limit of Image of Sequence",
"Definition:Sequence"
... |
proofwiki-1581 | Convergent Subsequence in Closed Interval | Let $\closedint a b$ be a closed real interval.
Then every sequence of points of $\closedint a b$ contains a subsequence which converges to a point in $\closedint a b$. | Let $\sequence {x_n}$ be a sequence in $\closedint a b$.
Since $\closedint a b$ is bounded in $\R$, it follows that $\sequence {x_n}$ is a bounded sequence.
By the Bolzano-Weierstrass Theorem, $\sequence {x_n}$ has a subsequence $\sequence {x_{n_r} }$ which is convergent.
Suppose $x_{n_r} \to l$ as $n \to \infty$.
Sinc... | Let $\closedint a b$ be a [[Definition:Closed Real Interval|closed real interval]].
Then every [[Definition:Real Sequence|sequence]] of points of $\closedint a b$ contains a [[Definition:Subsequence|subsequence]] which [[Definition:Convergent Real Sequence|converges]] to a point in $\closedint a b$. | Let $\sequence {x_n}$ be a [[Definition:Sequence|sequence]] in $\closedint a b$.
Since $\closedint a b$ is [[Definition:Bounded Subset of Real Numbers|bounded in $\R$]], it follows that $\sequence {x_n}$ is a [[Definition:Bounded Sequence|bounded sequence]].
By the [[Bolzano-Weierstrass Theorem]], $\sequence {x_n}$ h... | Convergent Subsequence in Closed Interval | https://proofwiki.org/wiki/Convergent_Subsequence_in_Closed_Interval | https://proofwiki.org/wiki/Convergent_Subsequence_in_Closed_Interval | [
"Analysis",
"Convergence",
"Limits of Sequences"
] | [
"Definition:Real Interval/Closed",
"Definition:Real Sequence",
"Definition:Subsequence",
"Definition:Convergent Sequence/Real Numbers"
] | [
"Definition:Sequence",
"Definition:Bounded Set/Real Numbers",
"Definition:Bounded Sequence",
"Bolzano-Weierstrass Theorem",
"Definition:Subsequence",
"Definition:Convergent Sequence",
"Lower and Upper Bounds for Sequences",
"Definition:Convergent Sequence"
] |
proofwiki-1582 | Max and Min of Function on Closed Real Interval | Let $f$ be a real function which is continuous on the closed interval $\closedint a b$.
Then $f$ reaches a maximum and a minimum on $\closedint a b$. | From Image of Closed Real Interval is Bounded, we have that $f$ is bounded on $\closedint a b$.
Let $d$ be the supremum of $f$ on $\closedint a b$.
Consider a sequence $\sequence {x_n}$ in $\closedint a b$ such that $\size {\map f {x_n} } \to d$ as $n \to \infty$.
From {{Corollary|Limit of Sequence to Zero Distance Poi... | Let $f$ be a [[Definition:Real Function|real function]] which is [[Definition:Continuous on Interval|continuous]] on the [[Definition:Closed Real Interval|closed interval]] $\closedint a b$.
Then $f$ reaches a [[Definition:Maximum Value|maximum]] and a [[Definition:Minimum Value|minimum]] on $\closedint a b$. | From [[Image of Closed Real Interval is Bounded]], we have that $f$ is [[Definition:Bounded Mapping|bounded]] on $\closedint a b$.
Let $d$ be the [[Definition:Supremum of Real-Valued Function|supremum]] of $f$ on $\closedint a b$.
Consider a [[Definition:Sequence|sequence]] $\sequence {x_n}$ in $\closedint a b$ such ... | Max and Min of Function on Closed Real Interval/Proof 1 | https://proofwiki.org/wiki/Max_and_Min_of_Function_on_Closed_Real_Interval | https://proofwiki.org/wiki/Max_and_Min_of_Function_on_Closed_Real_Interval/Proof_1 | [
"Real Intervals",
"Max and Min of Function on Closed Real Interval"
] | [
"Definition:Real Function",
"Definition:Continuous Real Function/Interval",
"Definition:Real Interval/Closed",
"Definition:Maximum Value of Real Function/Absolute",
"Definition:Minimum Value of Real Function/Absolute"
] | [
"Image of Closed Real Interval is Bounded",
"Definition:Bounded Mapping",
"Definition:Supremum of Mapping/Real-Valued Function",
"Definition:Sequence",
"Definition:Real Interval/Closed",
"Convergent Subsequence in Closed Interval",
"Definition:Subsequence",
"Definition:Convergent Sequence",
"Definit... |
proofwiki-1583 | Max and Min of Function on Closed Real Interval | Let $f$ be a real function which is continuous on the closed interval $\closedint a b$.
Then $f$ reaches a maximum and a minimum on $\closedint a b$. | This is an instance of the Extreme Value Theorem.
$\closedint a b$ is a compact subset of a metric space from Real Number Line is Metric Space.
$\R$ itself is a normed vector space.
{{MissingLinks|Find the reference to $\R$ being a normed vector space.}}
Hence the result.
{{qed}} | Let $f$ be a [[Definition:Real Function|real function]] which is [[Definition:Continuous on Interval|continuous]] on the [[Definition:Closed Real Interval|closed interval]] $\closedint a b$.
Then $f$ reaches a [[Definition:Maximum Value|maximum]] and a [[Definition:Minimum Value|minimum]] on $\closedint a b$. | This is an instance of the [[Extreme Value Theorem]].
$\closedint a b$ is a [[Definition:Compact Subset of Metric Space|compact subset]] of a [[Definition:Metric Space|metric space]] from [[Real Number Line is Metric Space]].
$\R$ itself is a [[Definition:Normed Vector Space|normed vector space]].
{{MissingLinks|Fin... | Max and Min of Function on Closed Real Interval/Proof 2 | https://proofwiki.org/wiki/Max_and_Min_of_Function_on_Closed_Real_Interval | https://proofwiki.org/wiki/Max_and_Min_of_Function_on_Closed_Real_Interval/Proof_2 | [
"Real Intervals",
"Max and Min of Function on Closed Real Interval"
] | [
"Definition:Real Function",
"Definition:Continuous Real Function/Interval",
"Definition:Real Interval/Closed",
"Definition:Maximum Value of Real Function/Absolute",
"Definition:Minimum Value of Real Function/Absolute"
] | [
"Extreme Value Theorem",
"Definition:Compact Subset of Metric Space",
"Definition:Metric Space",
"Real Number Line is Metric Space",
"Definition:Normed Vector Space",
"Definition:Normed Vector Space"
] |
proofwiki-1584 | Continuous Image of Closed Real Interval is Closed Real Interval | Let $f$ be a real function which is continuous on the closed interval $\closedint a b$.
Then the image of $\closedint a b$ under $f$ is also a closed interval. | Let $I = \closedint a b$.
Let $J = f \sqbrk I$.
From Image of Real Interval under Continuous Real Function is Real Interval, $J$ is an interval.
From Image of Closed Real Interval is Bounded, $J$ is bounded.
From Max and Min of Function on Closed Real Interval, $J$ includes its end points.
Hence the result.
{{qed}} | Let $f$ be a [[Definition:Real Function|real function]] which is [[Definition:Continuous on Interval|continuous]] on the [[Definition:Closed Real Interval|closed interval]] $\closedint a b$.
Then the [[Definition:Image of Subset under Mapping|image]] of $\closedint a b$ under $f$ is also a [[Definition:Closed Real Int... | Let $I = \closedint a b$.
Let $J = f \sqbrk I$.
From [[Image of Real Interval under Continuous Real Function is Real Interval]], $J$ is an [[Definition:Real Interval|interval]].
From [[Image of Closed Real Interval is Bounded]], $J$ is [[Definition:Bounded Mapping|bounded]].
From [[Max and Min of Function on Closed... | Continuous Image of Closed Real Interval is Closed Real Interval | https://proofwiki.org/wiki/Continuous_Image_of_Closed_Real_Interval_is_Closed_Real_Interval | https://proofwiki.org/wiki/Continuous_Image_of_Closed_Real_Interval_is_Closed_Real_Interval | [
"Continuous Real Functions",
"Real Intervals"
] | [
"Definition:Real Function",
"Definition:Continuous Real Function/Interval",
"Definition:Real Interval/Closed",
"Definition:Image (Set Theory)/Mapping/Subset",
"Definition:Real Interval/Closed"
] | [
"Image of Real Interval under Continuous Real Function is Real Interval",
"Definition:Real Interval",
"Image of Closed Real Interval is Bounded",
"Definition:Bounded Mapping",
"Max and Min of Function on Closed Real Interval",
"Definition:Real Interval/Endpoints"
] |
proofwiki-1585 | Retraction Theorem | Let $M$ be a compact manifold with boundary $\partial M$.
Then there is no smooth mapping $f: M \to \partial M$ such that $\partial f: \partial M \to \partial M$ is the identity. | {{MissingLinks|one-dimensional, for a start}}
{{AimForCont}} such a smooth mapping exists.
By the Morse-Sard Theorem, there exists a regular value $x \in \partial M$.
By the Preimage Theorem:
:$\map {f^{-1} } x$ is a submanifold of $M$ with boundary.
We have that the codimension of $\map {f^{-1} } x$ in $M$ equals the ... | Let $M$ be a [[Definition:Compact Topological Space|compact]] [[Definition:Topological Manifold|manifold]] with [[Definition:Boundary (Topology)|boundary]] $\partial M$.
Then there is no [[Definition:Smooth Mapping|smooth mapping]] $f: M \to \partial M$ such that $\partial f: \partial M \to \partial M$ is the [[Defin... | {{MissingLinks|one-dimensional, for a start}}
{{AimForCont}} such a [[Definition:Smooth Mapping|smooth mapping]] exists.
By the [[Morse-Sard Theorem]], there exists a [[Definition:Regular Value|regular value]] $x \in \partial M$.
By the [[Preimage Theorem]]:
:$\map {f^{-1} } x$ is a [[Definition:Submanifold|submanif... | Retraction Theorem | https://proofwiki.org/wiki/Retraction_Theorem | https://proofwiki.org/wiki/Retraction_Theorem | [
"Algebraic Topology",
"Named Theorems"
] | [
"Definition:Compact Topological Space",
"Definition:Topological Manifold",
"Definition:Boundary (Topology)",
"Definition:Smooth Mapping",
"Definition:Identity Mapping"
] | [
"Definition:Smooth Mapping",
"Morse-Sard Theorem",
"Definition:Regular Value",
"Preimage Theorem",
"Definition:Submanifold",
"Definition:Codimension",
"Definition:Codimension",
"Definition:Compact Topological Space",
"Definition:Identity Mapping",
"Classification of Compact One-Manifolds",
"Cate... |
proofwiki-1586 | Classification of Compact One-Manifolds | Every compact connected one-dimensional manifold is diffeomorphic to either a circle or a closed interval. | === Lemma 1 ===
{{:Classification of Compact One-Manifolds/Lemma 1}}{{qed|lemma}}
Let $f$ be a Morse function on a one-manifold $X$.
Let $S$ be the union of the critical points of $f$ and $\partial X$.
As $S$ is finite, $X \setminus S$ consists of a finite number of one-manifolds, $L_1, L_2, \cdots, L_n$. | Every [[Definition:Compact Topological Space|compact]] [[Definition:Connected Topological Space|connected]] one-[[Definition:Dimension (Topology)|dimensional]] [[Definition:Topological Manifold|manifold]] is [[Definition:Diffeomorphism|diffeomorphic]] to either a [[Definition:Circle|circle]] or a [[Definition:Closed In... | === [[Classification of Compact One-Manifolds/Lemma 1|Lemma 1]] ===
{{:Classification of Compact One-Manifolds/Lemma 1}}{{qed|lemma}}
Let $f$ be a [[Definition:Morse Function|Morse function]] on a one-[[Definition:Topological Manifold|manifold]] $X$.
Let $S$ be the [[Definition:Set Union|union]] of the [[Definition:... | Classification of Compact One-Manifolds | https://proofwiki.org/wiki/Classification_of_Compact_One-Manifolds | https://proofwiki.org/wiki/Classification_of_Compact_One-Manifolds | [
"Topological Manifolds",
"Classification of Compact One-Manifolds"
] | [
"Definition:Compact Topological Space",
"Definition:Connected Topological Space",
"Definition:Dimension (Topology)",
"Definition:Topological Manifold",
"Definition:Diffeomorphism",
"Definition:Circle",
"Definition:Interval/Ordered Set/Closed"
] | [
"Classification of Compact One-Manifolds/Lemma 1",
"Definition:Morse Function",
"Definition:Topological Manifold",
"Definition:Set Union",
"Definition:Critical Point (Topology)",
"Definition:Finite Set",
"Definition:Finite Cardinal",
"Definition:Topological Manifold"
] |
proofwiki-1587 | Differentiable Function is Continuous | Let $f$ be a real function defined on an interval $I$.
Let $x_0 \in I$ such that $f$ is differentiable at $x_0$.
Then $f$ is continuous at $x_0$. | Let $\map {N_r} 0$ denote the $r$-neighborhood of $0$ in $\C$.
By the Epsilon-Function Complex Differentiability Condition, it follows that there exists $r \in \R_{>0}$ such that for all $h \in \map {N_r} 0 \setminus \set 0$:
:$(1): \quad \map f {a + h} = \map f a + h \paren {\map {f'} a + \map \epsilon h}$
where $\eps... | Let $f$ be a [[Definition:Real Function|real function]] defined on an [[Definition:Real Interval|interval]] $I$.
Let $x_0 \in I$ such that $f$ is [[Definition:Differentiable Real Function at Point|differentiable]] at $x_0$.
Then $f$ is [[Definition:Continuous Real Function at Point|continuous]] at $x_0$. | Let $\map {N_r} 0$ denote the [[Definition:Neighborhood (Complex Analysis)|$r$-neighborhood]] of $0$ in $\C$.
By the [[Epsilon-Function Complex Differentiability Condition]], it follows that there exists $r \in \R_{>0}$ such that for all $h \in \map {N_r} 0 \setminus \set 0$:
:$(1): \quad \map f {a + h} = \map f a + ... | Complex-Differentiable Function is Continuous/Proof 1 | https://proofwiki.org/wiki/Differentiable_Function_is_Continuous | https://proofwiki.org/wiki/Complex-Differentiable_Function_is_Continuous/Proof_1 | [
"Differentiable Real Functions",
"Continuous Real Functions"
] | [
"Definition:Real Function",
"Definition:Real Interval",
"Definition:Differentiable Mapping/Real Function/Point",
"Definition:Continuous Real Function/Point"
] | [
"Definition:Neighborhood (Complex Analysis)",
"Epsilon-Function Differentiability Condition/Complex Case",
"Definition:Complex Function",
"Combination Theorem for Limits of Functions/Complex",
"Definition:Continuous Complex Function/Using Limit",
"Definition:Continuous Complex Function"
] |
proofwiki-1588 | Differentiable Function is Continuous | Let $f$ be a real function defined on an interval $I$.
Let $x_0 \in I$ such that $f$ is differentiable at $x_0$.
Then $f$ is continuous at $x_0$. | For each $z \in D$:
{{begin-eqn}}
{{eqn | l = \lim_{w \mathop \to z} \map f w
| r = \map f z + \lim_{w \mathop \to z} \paren {\map f w - \map f z}
| c = Sum Rule for Limits of Complex Functions
}}
{{eqn | r = \map f z + \lim_{w \mathop \to z} \paren {\frac {\map f w - \map f z} {w - z} \paren {w - z} }
}}
{... | Let $f$ be a [[Definition:Real Function|real function]] defined on an [[Definition:Real Interval|interval]] $I$.
Let $x_0 \in I$ such that $f$ is [[Definition:Differentiable Real Function at Point|differentiable]] at $x_0$.
Then $f$ is [[Definition:Continuous Real Function at Point|continuous]] at $x_0$. | For each $z \in D$:
{{begin-eqn}}
{{eqn | l = \lim_{w \mathop \to z} \map f w
| r = \map f z + \lim_{w \mathop \to z} \paren {\map f w - \map f z}
| c = [[Sum Rule for Limits of Complex Functions]]
}}
{{eqn | r = \map f z + \lim_{w \mathop \to z} \paren {\frac {\map f w - \map f z} {w - z} \paren {w - z} }
... | Complex-Differentiable Function is Continuous/Proof 2 | https://proofwiki.org/wiki/Differentiable_Function_is_Continuous | https://proofwiki.org/wiki/Complex-Differentiable_Function_is_Continuous/Proof_2 | [
"Differentiable Real Functions",
"Continuous Real Functions"
] | [
"Definition:Real Function",
"Definition:Real Interval",
"Definition:Differentiable Mapping/Real Function/Point",
"Definition:Continuous Real Function/Point"
] | [
"Combination Theorem for Limits of Functions/Complex/Sum Rule",
"Combination Theorem for Limits of Functions/Complex/Product Rule"
] |
proofwiki-1589 | Differentiable Function is Continuous | Let $f$ be a real function defined on an interval $I$.
Let $x_0 \in I$ such that $f$ is differentiable at $x_0$.
Then $f$ is continuous at $x_0$. | We have {{hypothesis}} that $\map {f'} {x_0}$ exists.
Let $x, x_0 \in I$ such that $x \ne x_0$.
Then:
{{begin-eqn}}
{{eqn | l = \map f x - \map f {x_0}
| r = \frac {\map f x - \map f {x_0} } {x - x_0} \cdot \paren {x - x_0}
| c =
}}
{{eqn | o = \to
| r = \map {f'} {x_0} \cdot 0
| c = as $x \to ... | Let $f$ be a [[Definition:Real Function|real function]] defined on an [[Definition:Real Interval|interval]] $I$.
Let $x_0 \in I$ such that $f$ is [[Definition:Differentiable Real Function at Point|differentiable]] at $x_0$.
Then $f$ is [[Definition:Continuous Real Function at Point|continuous]] at $x_0$. | We have {{hypothesis}} that $\map {f'} {x_0}$ exists.
Let $x, x_0 \in I$ such that $x \ne x_0$.
Then:
{{begin-eqn}}
{{eqn | l = \map f x - \map f {x_0}
| r = \frac {\map f x - \map f {x_0} } {x - x_0} \cdot \paren {x - x_0}
| c =
}}
{{eqn | o = \to
| r = \map {f'} {x_0} \cdot 0
| c = as $x \t... | Differentiable Function is Continuous | https://proofwiki.org/wiki/Differentiable_Function_is_Continuous | https://proofwiki.org/wiki/Differentiable_Function_is_Continuous | [
"Differentiable Real Functions",
"Continuous Real Functions"
] | [
"Definition:Real Function",
"Definition:Real Interval",
"Definition:Differentiable Mapping/Real Function/Point",
"Definition:Continuous Real Function/Point"
] | [
"Definition:Continuous Real Function/Point"
] |
proofwiki-1590 | Preimage Theorem | Let $y$ be a regular value of a smooth submersion $f: X \to Y$.
Then the preimage $\map {f^{-1} } y$ is a smooth submanifold of $X$, with $\dim \map {f^{-1} } y = \dim X - \dim Y$. | Let $k, l$ be natural numbers with $k \ge l$.
By the Local Submersion Theorem, there exists coordinates in some open neighborhoods of $x, y$ such that $\map f {x_1, x_2, \ldots, x_k} = \tuple {x_1, \ldots, x_l}$ and $y$ corresponds to $\tuple {0, \ldots, 0}$.
Let $V$ be that neighborhood of $x$.
Then $\map {f^{-1} } y ... | Let $y$ be a [[Definition:Regular Value|regular value]] of a [[Definition:Smooth Mapping|smooth]] [[Definition:Submersion|submersion]] $f: X \to Y$.
Then the [[Definition:Preimage of Element under Mapping|preimage]] $\map {f^{-1} } y$ is a smooth [[Definition:Submanifold|submanifold]] of $X$, with $\dim \map {f^{-1} }... | Let $k, l$ be [[Definition:Natural Numbers|natural numbers]] with $k \ge l$.
By the [[Local Submersion Theorem]], there exists coordinates in some open neighborhoods of $x, y$ such that $\map f {x_1, x_2, \ldots, x_k} = \tuple {x_1, \ldots, x_l}$ and $y$ corresponds to $\tuple {0, \ldots, 0}$.
Let $V$ be that [[Defin... | Preimage Theorem | https://proofwiki.org/wiki/Preimage_Theorem | https://proofwiki.org/wiki/Preimage_Theorem | [
"Named Theorems",
"Smooth Manifolds"
] | [
"Definition:Regular Value",
"Definition:Smooth Mapping",
"Definition:Submersion",
"Definition:Preimage/Mapping/Element",
"Definition:Submanifold"
] | [
"Definition:Natural Numbers",
"Local Submersion Theorem",
"Definition:Neighborhood (Topology)",
"Definition:Mapping",
"Definition:Coordinate System",
"Definition:Relatively Open",
"Definition:Diffeomorphism",
"Definition:Euclidean Space",
"Definition:Surjection",
"Definition:Tangent Space",
"Def... |
proofwiki-1591 | Forward-Backward Induction | Let $P$ be a propositional function on the natural numbers $\N$.
Suppose that:
:$(1): \quad \forall n \in \N: \map P {2^n}$ holds.
:$(2): \quad \map P n \implies \map P {n - 1}$.
Then $\map P n$ holds for all $\forall n \in \N$.
The proof technique based on this result is called '''forward-backward induction'''. | {{AimForCont}} $\exists k \in \N$ such that $\map P k$ is false.
From Power of Real Number greater than One is Unbounded Above:
:$\set {2^n: n \in \N}$ is unbounded above.
Therefore we can find:
:$M = 2^N > k$
Now let us create the set:
:$S = \set {n \in \N: n < M, \map P n \text { is false} }$
As $k < M$ and $\map P k... | Let $P$ be a [[Definition:Propositional Function|propositional function]] on the [[Definition:Natural Numbers|natural numbers]] $\N$.
Suppose that:
:$(1): \quad \forall n \in \N: \map P {2^n}$ holds.
:$(2): \quad \map P n \implies \map P {n - 1}$.
Then $\map P n$ holds for all $\forall n \in \N$.
The proof techniqu... | {{AimForCont}} $\exists k \in \N$ such that $\map P k$ is [[Definition:False|false]].
From [[Power of Real Number greater than One is Unbounded Above]]:
:$\set {2^n: n \in \N}$ is [[Definition:Bounded Above Set|unbounded above]].
Therefore we can find:
:$M = 2^N > k$
Now let us create the [[Definition:Set|set]]:
:$S... | Forward-Backward Induction | https://proofwiki.org/wiki/Forward-Backward_Induction | https://proofwiki.org/wiki/Forward-Backward_Induction | [
"Forward-Backward Induction",
"Number Theory",
"Named Theorems",
"Mathematical Induction",
"Proof Techniques"
] | [
"Definition:Propositional Function",
"Definition:Natural Numbers",
"Forward-Backward Induction"
] | [
"Definition:False",
"Power of Real Number greater than One is Unbounded Above",
"Definition:Bounded Above Set",
"Definition:Set",
"Definition:False",
"Definition:Bounded Above Set",
"Set of Integers Bounded Above by Integer has Greatest Element",
"Definition:Greatest Element",
"Definition:Contradict... |
proofwiki-1592 | Inequalities Concerning Roots | Let $\closedint X Y$ be a closed real interval such that $0 < X \le Y$.
Let $x, y \in \closedint X Y$.
Then:
:$\forall n \in \N_{> 0}: X Y^{1/n} \size {x - y} \le n X Y \size {x^{1/n} - y^{1/n} } \le Y X^{1/n} \size {x - y}$ | From Difference of Two Powers:
:$\ds a^n - b^n = \paren {a - b} \paren {a^{n - 1} + a^{n - 2} b + a^{n - 3} b^2 + \dotsb + a b^{n - 2} + b^{n - 1} } = \paren {a - b} \sum_{j \mathop = 0}^{n - 1} a^{n - j - 1} b^j$
Let $a > b > 0$.
Then:
:$\dfrac {a^n - b^n} {a - b} = a^{n - 1} + a^{n - 2} b + a^{n - 3} b^2 + \dotsb + a... | Let $\closedint X Y$ be a [[Definition:Closed Real Interval|closed real interval]] such that $0 < X \le Y$.
Let $x, y \in \closedint X Y$.
Then:
:$\forall n \in \N_{> 0}: X Y^{1/n} \size {x - y} \le n X Y \size {x^{1/n} - y^{1/n} } \le Y X^{1/n} \size {x - y}$ | From [[Difference of Two Powers]]:
:$\ds a^n - b^n = \paren {a - b} \paren {a^{n - 1} + a^{n - 2} b + a^{n - 3} b^2 + \dotsb + a b^{n - 2} + b^{n - 1} } = \paren {a - b} \sum_{j \mathop = 0}^{n - 1} a^{n - j - 1} b^j$
Let $a > b > 0$.
Then:
:$\dfrac {a^n - b^n} {a - b} = a^{n - 1} + a^{n - 2} b + a^{n - 3} b^2 + \dot... | Inequalities Concerning Roots | https://proofwiki.org/wiki/Inequalities_Concerning_Roots | https://proofwiki.org/wiki/Inequalities_Concerning_Roots | [
"Roots of Numbers",
"Inequalities"
] | [
"Definition:Real Interval/Closed"
] | [
"Difference of Two Powers",
"Definition:Reciprocal"
] |
proofwiki-1593 | Continuity of Root Function | Let $n \in \N_{>0}$ be a non-zero natural number.
Let $f: \hointr 0 \infty \to \R$ be the real function defined by $\map f x = x^{1/n}$.
Then $f$ is continuous at each $\xi > 0$ and continuous on the right at $\xi = 0$. | First suppose that $\xi > 0$.
Let $X, Y \in \R$ such that $0 < X < \xi < Y$.
Let $x \in \R$ such that $X < x < Y$.
From Inequalities Concerning Roots:
:$X Y^{1/n} \ \size {x - \xi} \le n X Y \ \size {x^{1/n} - \xi^{1/n} } \le Y X^{1/n} \ \size {x - \xi}$
Thus:
:$\dfrac 1 {n Y} Y^{1/n} \ \size {x - \xi} \le \size {x^{1/... | Let $n \in \N_{>0}$ be a non-zero [[Definition:Natural Number|natural number]].
Let $f: \hointr 0 \infty \to \R$ be the [[Definition:Real Function|real function]] defined by $\map f x = x^{1/n}$.
Then $f$ is [[Definition:Continuous Real Function at Point|continuous]] at each $\xi > 0$ and [[Definition:Right-Continuo... | First suppose that $\xi > 0$.
Let $X, Y \in \R$ such that $0 < X < \xi < Y$.
Let $x \in \R$ such that $X < x < Y$.
From [[Inequalities Concerning Roots]]:
:$X Y^{1/n} \ \size {x - \xi} \le n X Y \ \size {x^{1/n} - \xi^{1/n} } \le Y X^{1/n} \ \size {x - \xi}$
Thus:
:$\dfrac 1 {n Y} Y^{1/n} \ \size {x - \xi} \le \siz... | Continuity of Root Function | https://proofwiki.org/wiki/Continuity_of_Root_Function | https://proofwiki.org/wiki/Continuity_of_Root_Function | [
"Roots of Numbers",
"Continuous Functions"
] | [
"Definition:Natural Numbers",
"Definition:Real Function",
"Definition:Continuous Real Function/Point",
"Definition:Continuous Real Function/Right-Continuous"
] | [
"Inequalities Concerning Roots",
"Squeeze Theorem"
] |
proofwiki-1594 | Weak Whitney Immersion Theorem | Every $k$-dimensional manifold $X$ admits a one-to-one immersion in $\R^{2 k + 1}$.
{{MissingLinks|$k$-dimensional}} | Let $M > 2 k + 1$ be a natural number such that $f: X \to \R^M$ is an injective immersion.
Define a map $h: X \times X \times \R \to \R^M$ by:
:$\map h {x, y, t} = \map t {\map f x - \map f y}$
Define a map $g: \map T X \to \R^M$ by:
:$\map g {x, v} = \map {\d f_x} v$
where $\map T X$ is the tangent bundle of $X$.
Sinc... | Every $k$-dimensional [[Definition:Topological Manifold|manifold]] $X$ admits a [[Definition:One-to-One Relation|one-to-one]] [[Definition:Immersion|immersion]] in $\R^{2 k + 1}$.
{{MissingLinks|$k$-dimensional}} | Let $M > 2 k + 1$ be a [[Definition:Natural Numbers|natural number]] such that $f: X \to \R^M$ is an [[Definition:Injection|injective]] [[Definition:Immersion|immersion]].
Define a map $h: X \times X \times \R \to \R^M$ by:
:$\map h {x, y, t} = \map t {\map f x - \map f y}$
Define a map $g: \map T X \to \R^M$ by:
:$\... | Weak Whitney Immersion Theorem | https://proofwiki.org/wiki/Weak_Whitney_Immersion_Theorem | https://proofwiki.org/wiki/Weak_Whitney_Immersion_Theorem | [
"Topological Manifolds"
] | [
"Definition:Topological Manifold",
"Definition:One-to-One Relation",
"Definition:Immersion"
] | [
"Definition:Natural Numbers",
"Definition:Injection",
"Definition:Immersion",
"Definition:Tangent Bundle",
"Morse-Sard Theorem",
"Definition:Injection",
"Proof by Contradiction",
"Definition:Injection",
"Definition:Tangent Space"
] |
proofwiki-1595 | P-adic Norm is Norm | The $p$-adic norm forms a norm on the rational numbers $\Q$. | Let $v_p$ be the $p$-adic valuation on the rational numbers.
Recall that the $p$-adic norm is defined as:
:$\forall q \in \Q: \norm q_p := \begin {cases} 0 & : q = 0 \\ p^{- \map {\nu_p} q} & : q \ne 0 \end {cases}$
We must show the following hold for all $x$, $y \in \Q$:
{{begin-axiom}}
{{axiom | n = \text N 1
... | The [[Definition:P-adic Norm|$p$-adic norm]] forms a [[Definition:Norm on Division Ring|norm]] on the [[Definition:Rational Number|rational numbers]] $\Q$. | Let $v_p$ be the [[Definition:P-adic Valuation|$p$-adic valuation]] on the [[Definition:Rational Number|rational numbers]].
Recall that the [[Definition:P-adic Norm|$p$-adic norm]] is defined as:
:$\forall q \in \Q: \norm q_p := \begin {cases} 0 & : q = 0 \\ p^{- \map {\nu_p} q} & : q \ne 0 \end {cases}$
We must sho... | P-adic Norm is Norm/Proof 1 | https://proofwiki.org/wiki/P-adic_Norm_is_Norm | https://proofwiki.org/wiki/P-adic_Norm_is_Norm/Proof_1 | [
"Examples of Norms",
"P-adic Norm is Norm"
] | [
"Definition:P-adic Norm",
"Definition:Norm/Division Ring",
"Definition:Rational Number"
] | [
"Definition:P-adic Valuation",
"Definition:Rational Number",
"Definition:P-adic Norm",
"Power of Positive Real Number is Positive/Real Number",
"Definition:P-adic Norm",
"Definition:P-adic Norm",
"P-adic Valuation is Valuation",
"Exponent Combination Laws/Product of Powers",
"Definition:P-adic Norm"... |
proofwiki-1596 | P-adic Norm is Norm | The $p$-adic norm forms a norm on the rational numbers $\Q$. | Recall that the $p$-adic norm is defined as:
:<nowiki>$\forall q \in \Q: \norm r_p := \begin {cases}
0 & : r = 0 \\
p^{- k} & : r \ne 0
\end {cases}$
</nowiki>
where:
:$r = p^k \dfrac m n$
and:
:$k, n \in \Z, m \in \Z_{\ne 0} : p \nmid m, n$
where $\nmid$ stands for "does not divide".
We must show that the norm axi... | The [[Definition:P-adic Norm|$p$-adic norm]] forms a [[Definition:Norm on Division Ring|norm]] on the [[Definition:Rational Number|rational numbers]] $\Q$. | Recall that the [[Definition:P-adic Norm|$p$-adic norm]] is defined as:
:<nowiki>$\forall q \in \Q: \norm r_p := \begin {cases}
0 & : r = 0 \\
p^{- k} & : r \ne 0
\end {cases}$
</nowiki>
where:
:$r = p^k \dfrac m n$
and:
:$k, n \in \Z, m \in \Z_{\ne 0} : p \nmid m, n$
where $\nmid$ stands for [[Symbols:Number ... | P-adic Norm is Norm/Proof 2 | https://proofwiki.org/wiki/P-adic_Norm_is_Norm | https://proofwiki.org/wiki/P-adic_Norm_is_Norm/Proof_2 | [
"Examples of Norms",
"P-adic Norm is Norm"
] | [
"Definition:P-adic Norm",
"Definition:Norm/Division Ring",
"Definition:Rational Number"
] | [
"Definition:P-adic Norm",
"Symbols:Number Theory/Does Not Divide",
"Axiom:Multiplicative Norm Axioms",
"Definition:P-adic Norm",
"Definition:P-adic Norm",
"Definition:P-adic Norm",
"Definition:Prime Number",
"Fundamental Theorem of Arithmetic",
"Axiom:Multiplicative Norm Axioms"
] |
proofwiki-1597 | Fundamental Theorem of Algebra | Every non-constant polynomial with coefficients in $\C$ has a root in $\C$.
That is, the field of complex numbers is algebraically closed. | Let $\map p z$ be a polynomial in $\C$:
:$\map p z = z^m + a_1 z^{m-1} + \cdots + a_m$
where not all of $a_1, \ldots, a_m$ are zero.
Define a homotopy:
:$\map {p_t} z = t \map p z + \left({1-t}\right) z^m$
Then:
:$\dfrac {\map {p_t} z} {z^m} = 1 + t \paren {a_1 \dfrac 1 z + \cdots + a_m \dfrac 1 {z^m} }$
The terms in t... | Every non-[[Definition:Constant Polynomial|constant]] [[Definition:Polynomial|polynomial]] with [[Definition:Polynomial Coefficient|coefficients]] in $\C$ has a [[Definition:Root of Polynomial|root]] in $\C$.
That is, the [[Definition:Complex Number|field of complex numbers]] is [[Definition:Algebraically Closed Field... | Let $\map p z$ be a [[Definition:Polynomial|polynomial]] in $\C$:
:$\map p z = z^m + a_1 z^{m-1} + \cdots + a_m$
where not all of $a_1, \ldots, a_m$ are zero.
Define a [[Definition:Free Homotopy|homotopy]]:
:$\map {p_t} z = t \map p z + \left({1-t}\right) z^m$
Then:
:$\dfrac {\map {p_t} z} {z^m} = 1 + t \paren {a_1 \... | Fundamental Theorem of Algebra/Proof 1 | https://proofwiki.org/wiki/Fundamental_Theorem_of_Algebra | https://proofwiki.org/wiki/Fundamental_Theorem_of_Algebra/Proof_1 | [
"Fundamental Theorem of Algebra",
"Polynomial Theory",
"Algebra",
"Analysis",
"Fundamental Theorems"
] | [
"Definition:Constant Polynomial",
"Definition:Polynomial",
"Definition:Coefficient of Polynomial",
"Definition:Root of Polynomial",
"Definition:Complex Number",
"Definition:Algebraically Closed Field"
] | [
"Definition:Polynomial",
"Definition:Homotopy/Free",
"Definition:Well-Defined/Mapping",
"Definition:Complex Number",
"Definition:Homotopy/Free",
"Definition:Constant Polynomial",
"Extendability Theorem for Intersection Numbers"
] |
proofwiki-1598 | Fundamental Theorem of Algebra | Every non-constant polynomial with coefficients in $\C$ has a root in $\C$.
That is, the field of complex numbers is algebraically closed. | Let $\map P z = a_n z^n + \dots + a_1 z + a_0, \ a_n \ne 0$.
{{AimForCont}} that $\map P z$ is not zero for any $z \in \C$.
It follows that $1 / \map P z$ must be entire; and is also bounded in the complex plane.
In order to see that it is indeed bounded, we recall that $\exists R \in \R_{>0}$ such that:
$\cmod {\df... | Every non-[[Definition:Constant Polynomial|constant]] [[Definition:Polynomial|polynomial]] with [[Definition:Polynomial Coefficient|coefficients]] in $\C$ has a [[Definition:Root of Polynomial|root]] in $\C$.
That is, the [[Definition:Complex Number|field of complex numbers]] is [[Definition:Algebraically Closed Field... | Let $\map P z = a_n z^n + \dots + a_1 z + a_0, \ a_n \ne 0$.
{{AimForCont}} that $\map P z$ is not zero for any $z \in \C$.
It follows that $1 / \map P z$ must be [[Definition:Entire Function|entire]]; and is also [[Definition:Bounded Complex-Valued Function|bounded]] in the complex plane.
In order to see that it is... | Fundamental Theorem of Algebra/Proof 2 | https://proofwiki.org/wiki/Fundamental_Theorem_of_Algebra | https://proofwiki.org/wiki/Fundamental_Theorem_of_Algebra/Proof_2 | [
"Fundamental Theorem of Algebra",
"Polynomial Theory",
"Algebra",
"Analysis",
"Fundamental Theorems"
] | [
"Definition:Constant Polynomial",
"Definition:Polynomial",
"Definition:Coefficient of Polynomial",
"Definition:Root of Polynomial",
"Definition:Complex Number",
"Definition:Algebraically Closed Field"
] | [
"Definition:Entire Function",
"Definition:Bounded Mapping/Complex-Valued",
"Liouville's Theorem (Complex Analysis)",
"Definition:Contradiction"
] |
proofwiki-1599 | Fundamental Theorem of Algebra | Every non-constant polynomial with coefficients in $\C$ has a root in $\C$.
That is, the field of complex numbers is algebraically closed. | Let $p: \C \to \C$ be a complex, non-constant polynomial.
{{AimForCont}} that $\map p z \ne 0$ for all $z \in \C$.
Now consider the closed contour integral:
:$\ds \oint \limits_{\gamma_R} \frac 1 {z \cdot \map p z} \rd z$
where $\gamma_R$ is a circle with radius $R$ around the origin.
By Derivative of Complex Polynomia... | Every non-[[Definition:Constant Polynomial|constant]] [[Definition:Polynomial|polynomial]] with [[Definition:Polynomial Coefficient|coefficients]] in $\C$ has a [[Definition:Root of Polynomial|root]] in $\C$.
That is, the [[Definition:Complex Number|field of complex numbers]] is [[Definition:Algebraically Closed Field... | Let $p: \C \to \C$ be a [[Definition:Complex Number|complex]], non-[[Definition:Constant Polynomial|constant]] [[Definition:Polynomial|polynomial]].
{{AimForCont}} that $\map p z \ne 0$ for all $z \in \C$.
Now consider the [[Definition:Closed Complex Contour Integral|closed contour integral]]:
:$\ds \oint \limits_{\g... | Fundamental Theorem of Algebra/Proof 3 | https://proofwiki.org/wiki/Fundamental_Theorem_of_Algebra | https://proofwiki.org/wiki/Fundamental_Theorem_of_Algebra/Proof_3 | [
"Fundamental Theorem of Algebra",
"Polynomial Theory",
"Algebra",
"Analysis",
"Fundamental Theorems"
] | [
"Definition:Constant Polynomial",
"Definition:Polynomial",
"Definition:Coefficient of Polynomial",
"Definition:Root of Polynomial",
"Definition:Complex Number",
"Definition:Algebraically Closed Field"
] | [
"Definition:Complex Number",
"Definition:Constant Polynomial",
"Definition:Polynomial",
"Definition:Contour Integral/Complex/Closed",
"Definition:Circle",
"Definition:Circle/Radius",
"Definition:Coordinate System/Origin",
"Derivative of Complex Polynomial",
"Definition:Holomorphic Function",
"Defi... |
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