Here is an outline of a proof.
We may as well prove this statement: Every bounded sequence contains a convergent subsequence.
Lemma 1, Every sequence, $\displaystyle \left(x_n\right)$, contains a monotone subsequence.
Easy proof: Let $\displaystyle S=\{n:~\forall j>n,~x_j>x_n\}$. Two cases: $\displaystyle S$ is infinite or finite.
In the first case the subsequence is increasing.
In the second case the subsequence is non-increasing.
Lemma 2, Every bounded monotone sequence converges.