Arzela-Ascoli theorem

For a compact Hausdorff space $K$, a subset of $C(K)$ is relatively compact in the uniform norm exactly when it is equicontinuous and pointwise bounded. On a compact metric space, this says that every uniformly bounded equicontinuous sequence has a uniformly convergent subsequence.

A subset of a topological space is relatively compact when its closure is compact.

If successive subsequences converge uniformly on an exhaustion $K_1\subset K_2\subset \cdots$, the diagonal sequence converges uniformly on every fixed $K_m$, hence locally uniformly on their union.

For a compactly supported nonzero function $\psi$, the translates $f_n(x)=\psi (x-n)$ converge uniformly to zero on every compact subset of $\mathbb{R}$, while $\Vert f_n{\Vert }_{\infty }=\Vert \psi {\Vert }_{\infty }$ prevents uniform convergence on all of $\mathbb{R}$.