Bounded Sequence in Euclidean Space has Convergent Subsequence/Proof 2

Proof
Let the range of $\sequence {x_i}$ be $S$.

By Closure of Bounded Subset of Metric Space is Bounded $\map \cl S$ is bounded in $\R^n$.

By Topological Closure is Closed, $\map \cl S$ 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.