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 Euclidean metric and is therefore a metric space.

A Metric Space is First-Countable.

The above conditions satisfy the hypotheses of Compact First-Countable Space is Sequentially Compact.

Therefore $\langle x_i \rangle$ has a convergent subsequence.