Bounded Sequence in Euclidean Space has Convergent Subsequence/Proof 2

Theorem

Let $\left\langle{x_i}\right\rangle_{i \in \N}$ be a bounded sequence in the Euclidean space $\R^n$.

Then some subsequence of $\left\langle{x_i}\right\rangle_{i \in \N}$ converges to a limit.

Proof

Let the range of $\left\langle{x_i}\right\rangle$ be $S$.

By Closure of Bounded Subset of Metric Space is Bounded $\operatorname{cl} \left({S}\right)$ is bounded in $\R^n$.

By Topological Closure is Closed, $\operatorname{cl} \left({S}\right)$ is closed in $\R^n$.

By the Heine-Borel Theorem, $S$ is compact.

The result follows from Compact Subspace of Metric Space is Sequentially Compact in Itself.

$\blacksquare$