# Explanation of the Bolzano-Weierstrass theorem proof

1. Oct 7, 2012

### BifSlamkovich

1. The problem statement, all variables and given/known data

Every bounded sequence has a convergent subsequence.

2. Relevant equations

Suppose that closed intervals I_0 $\supset$ I_1$\supset$ ... $\supset$ I_m and natural numbers n$_{1}$ < n$_{2}$ <
... < n$_{m}$ have been chosen such that for each 0 $\leq$ k $\leq$ m,
(2)
|I$_{k}$| = $b-a/2^{k}$, x$_{n}_{k}$$\in$I$_{k}$n and x$_{n}$ $\in$ I$_{k}$ for infinitely many n.

3. The attempt at a solution