Difference between revisions of "Bolzano-Weierstrass Theorem"

Latest revision as of 16:18, 12 June 2022

The Bolzano-Weierstrass Theorem is a result in analysis that states that every bounded sequence of real numbers $(a_n)$ contains a convergent subsequence.

Proof: Since $(a_n)$ is assumed to be bounded we have $|a_n|\le M$ . Bisect the closed interval $[-M,M]$ into two intervals $[-M,0]$ and $[0,M]$ . Let $I_1=[-M,0]$ . Take some point $a_{n_1}\in I_1$ . Bisect $I_1$ into two new intervals, and label the rightmost interval $I_2$ . Since there are infinite points in $I_2$ we can pick some $a_{n_2}\in I_2$ , and continue this process by picking some $a_{n_k}\in I_k$ . We show that the sequence $(a_{n_k})$ is convergent. Consider the chain

$\ldots\subseteq I_k\subseteq I_{k-1}\subseteq\ldots\subseteq I_2\subseteq I_1.$

By the Nested Interval Property we know that there is some $x\in\mathbb{R}$ contained in each interval. We claim $\lim a_{n_k}=x$ . Let $\epsilon>0$ be arbitrary. The length $\mathcal{L}(I_k)$ for each $I_k$ is, by construction, $\mathcal{L}(I_k)=M(1/2)^{k-1}$ which converges to $0$ . Choose $N$ such that for each $k\ge N$ that $\mathcal{L}(I_k)<\epsilon$ . Since $a_{n_k}$ and $x$ are contained in each interval, it follows that $|a_{n_k}-x|<\epsilon$ .

@@ Line 1: / Line 1: @@
-== Statement ==
+The '''Bolzano-Weierstrass Theorem''' is a result in [[analysis]] that states that every bounded [[sequence]] of real numbers <math>(a_n)</math> contains a convergent subsequence.
-Every bounded [[sequence]] of reals contains a convergent subsequence.
-== Proof ==
-{{solution}}
-== See also ==
+''Proof'': Since <math>(a_n)</math> is assumed to be bounded we have <math>|a_n|\le M</math>.  Bisect the closed interval <math>[-M,M]</math> into two intervals <math>[-M,0]</math> and <math>[0,M]</math>.  Let <math>I_1=[-M,0]</math>.  Take some point <math>a_{n_1}\in I_1</math>.  Bisect <math>I_1</math> into two new intervals, and label the rightmost interval <math>I_2</math>.  Since there are infinite points in <math>I_2</math> we can pick some <math>a_{n_2}\in I_2</math>, and continue this process by picking some <math>a_{n_k}\in I_k</math>.  We show that the sequence <math>(a_{n_k})</math> is convergent.  Consider the chain
+<center><cmath>\ldots\subseteq I_k\subseteq I_{k-1}\subseteq\ldots\subseteq I_2\subseteq I_1.</cmath></center>
+By the Nested Interval Property we know that there is some <math>x\in\mathbb{R}</math> contained in each interval.  We claim <math>\lim a_{n_k}=x</math>.  Let <math>\epsilon>0</math> be arbitrary.  The length <math>\mathcal{L}(I_k)</math> for each <math>I_k</math> is, by construction, <math>\mathcal{L}(I_k)=M(1/2)^{k-1}</math> which converges to <math>0</math>.  Choose <math>N</math> such that for each <math>k\ge N</math> that <math>\mathcal{L}(I_k)<\epsilon</math>.  Since <math>a_{n_k}</math> and <math>x</math> are contained in each interval, it follows that <math>|a_{n_k}-x|<\epsilon</math>.
-[[Category:Analysis]]
-[[Category:Theorems]]
-{{stub}}

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Difference between revisions of "Bolzano-Weierstrass Theorem"

Latest revision as of 16:18, 12 June 2022