Bounded Sequence in Euclidean Space has Convergent Subsequence/Proof 1

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

Because $\sequence {x_i}_{i \in \N}$ is bounded, we find $v \in \R^n$ and $\epsilon \in \R_{>0}$ such that:


 * $\map d {v, x_i} < \epsilon$

for all $i \in \N$.

Therefore, all $x_i$ are members of the closed $\epsilon$-ball $S = \map {B_\epsilon^-} v$.

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 $\struct {S, d \restriction_{S \times S}}$.

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

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