Closed, uniformly bounded, equicontinuous families on a compact space are compact in the uniform norm.Suggest a slogan
Let \(X\) be a compact Hausdorff space, and give \(C(X;\RR)\) the uniform norm \[ \norm{f}_{\infty,X}\coloneqq\sup_{x\in X}\abs{f(x)}, \] under which it is a Banach space. A subset \(A\subseteq C(X;\RR)\) is compact if and only if \(A\) is closed, uniformly bounded, and equicontinuous at every point of \(X\).
In particular, for \(X=[a,b]\subseteq\RR\), every uniformly bounded and uniformly equicontinuous sequence in \(C([a,b];\RR)\) has a uniformly convergent subsequence.