Bolzano-Weierstrass Theorem

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

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

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, the result follows.