# Bolzano-Weierstrass Theorem/Proof 1

## Theorem

Every bounded sequence of real numbers has a convergent subsequence.

## Proof

Let $\sequence {x_n}$ be a bounded sequence in $\R$.

By the Peak Point Lemma, $\sequence {x_n}$ has a monotone subsequence $\sequence {x_{n_r} }$.

Since $\sequence {x_n}$ is bounded, so is $\sequence {x_{n_r} }$.

Hence, by the Monotone Convergence Theorem (Real Analysis), the result follows.

$\blacksquare$

## Source of Name

This entry was named for Bernhard Bolzano and Karl Weierstrass.