Bounded Sequence in Euclidean Space has Convergent Subsequence

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
Denote with $d$ the Euclidean metric on $\R^n$.

Because $\left\langle{x_i}\right\rangle_{i \in \N}$ is bounded, we find $v \in \R^n$ and $M \in \R_{>0}$ such that:


 * $d \left({v, x_i}\right) < M$

for all $i \in \N$.

Therefore, all $x_i$ are members of the closed ball $S = B^- \left({v; M}\right)$.

By Closed Ball in Euclidean Space is Compact, $S$ is compact.

Thus $\left\langle{x_i}\right\rangle_{i \in \N}$ can be considered as a sequence in the compact metric space $\left({S, d \restriction_{S \times S}}\right)$.

By Compact Subspace of Metric Space is Sequentially Compact in Itself, $\left\langle{x_i}\right\rangle_{i \in \N}$ has a convergent subsequence in $S$.

In particular, since $S$ is a metric subspace of $\R^n$, it follows that $\left\langle{x_i}\right\rangle_{i \in \N}$ has a convergent subsequence in $\R^n$ as well.