Bolzano-Weierstrass Theorem/Proof 1

Theorem
Every bounded sequence of real numbers has a convergent subsequence.

Proof
Let $\left \langle {x_n} \right \rangle$ be a bounded sequence in $\R$.

By the Peak Point Lemma, $\left \langle {x_n} \right \rangle$ has a monotone subsequence $\left \langle {x_{n_r}} \right \rangle$.

Since $\left \langle {x_n} \right \rangle$ is bounded, so is $\left \langle {x_{n_r}} \right \rangle$.

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

Also see

 * Heine-Borel Theorem