Bounded Sequence in Euclidean Space has Convergent Subsequence/Proof 2

Theorem

Let $\sequence {x_i}_{i \mathop \in \N}$ be a bounded sequence in the Euclidean space $\R^n$.

Then some subsequence of $\sequence {x_i}_{i \mathop \in \N}$ converges to a limit.

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.

$\blacksquare$

Sources

• 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $7.2$: Sequential compactness: Corollary $7.2.7$