Bolzano-Weierstrass Theorem

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 $|a_{n_k}-x|<\epsilon$.