Arzelà–Ascoli theorem

In finite dimensions, Bolzano–Weierstrass says any bounded sequence has a convergent subsequence. In infinite-dimensional function spaces boundedness alone fails — sequences can be bounded yet escape to nowhere. Arzelà–Ascoli is the rescue: it pins down EXACTLY the extra condition (equicontinuity) that restores 'bounded implies a convergent subsequence' for continuous functions.

Theorem: let K be a compact metric space. A subset F of C(K) has compact closure (so every sequence in F has a uniformly convergent subsequence) if and only if F is uniformly bounded AND equicontinuous. In sequence form: a sequence of functions on a compact set that is uniformly bounded and equicontinuous has a subsequence converging uniformly to a continuous function.

The deep content is the characterization of COMPACTNESS in C(K): the relatively compact sets are precisely the bounded equicontinuous ones — boundedness keeps the functions from escaping vertically, equicontinuity keeps them from oscillating away. The theorem is the engine behind existence proofs throughout analysis: solutions of differential equations (Peano's theorem), the calculus of variations, normal families, and compactness of integral operators all lean on it.