Arzelà–Ascoli theorem

The Arzelà–Ascoli theorem is a fundamental result in mathematical analysis that characterizes when a family of functions has a uniformly convergent subsequence. Specifically, it states that if a set of functions defined on a closed interval is both uniformly bounded (they don't go off to infinity) and equicontinuous (their graphs don't wiggle too wildly), then there exists a sequence of functions within this set that converges uniformly to a continuous function. This theorem is crucial for understanding how sequences of functions behave and ensuring the existence of limits in various mathematical contexts.