Bounded Sequence in Euclidean Space has Convergent Subsequence

Theorem
Let $\langle x_i \rangle$ be a bounded sequence in Euclidean Space $\R^n$.

Then, some subsequence of $\langle x_i \rangle$ converges to a limit.

Proof
Because $\langle x_i \rangle$ is bounded, it is a subset of a compact subset of $\R^n$ by the Heine-Borel Theorem (Special Case). Moreover, Euclidean Space uses the Pythagorean Metric and is therefore a metric space. Every metric space is first-countable.

The above conditions satisfy the hypotheses of Sequences in a Compact Space Have a Convergent Subsequence. Therefore, we can conclude that $\langle x_i \rangle$ has a convergent subsequence.