 Compactness from uniformly vanishing sequence tails

Let $a_m\ge 0$ and $a_m\to 0$. The set $K=\{x=(x_m):x_m\in \mathbb{R},|x_m|\le a_m\}$ is a compact metric space for the supremum norm metric. Every member is a sequence tending to zero.

For any sequence in $K$, successively extract a convergent subsequence in each compact interval $[-a_m,a_m]$, then take diagonal indices. The coordinate limits define $x\in K$. For any $M$, coordinatewise differences in the tail are bounded by $2{\sup }_{m>M}{a}_m$, independently of the sequence index. This tends to zero as $M\to \infty$, while finitely many initial coordinates converge together. Hence the diagonal subsequence converges in the supremum norm. This proves sequential compactness, equivalent to compactness for metric spaces.