Bolzano-Weierstrass Theorem/Proof 1

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.

Also see

 * Heine-Borel Theorem