Bounded Sequence in Euclidean Space has Convergent Subsequence/Proof 2

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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 $\sequence {x_i}$ 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.

$\blacksquare$


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