Bolzano–Weierstrass theorem

This theorem is a promise about crowding. If you scatter infinitely many points but keep them all inside a finite stretch of the line, they have no room to spread out forever; somewhere they must bunch up. Out of any such bounded sequence you can always pull a subsequence that homes in on a single point.

Precisely: every bounded sequence of real numbers has at least one convergent subsequence. The limit of that subsequence is necessarily a point that the original sequence revisits arbitrarily closely infinitely often — a cluster point. The sequence itself need not converge; the theorem only promises a well-behaved subsequence hiding inside it.

Bolzano-Weierstrass is a cornerstone of analysis, equivalent in strength to the completeness of the reals, and it underlies compactness arguments throughout the subject — the proofs of the extreme value theorem, of sequential compactness of closed bounded sets, and of the Heine-Borel theorem all lean on it. A standard proof bisects the interval repeatedly, keeping a half containing infinitely many terms, producing nested intervals whose common point is the subsequential limit.