Bolzano-Weierstrass theorem

Every bounded real sequence has a convergent subsequence. One proof repeatedly bisects a closed interval containing infinitely many terms, chooses a nested half containing infinitely many terms, and then chooses indices increasingly from those halves. Their interval diameters tend to zero, so completeness gives convergence.

A diagonal subsequence argument successively extracts nested subsequences that converge in the first, first two, and then first $k$ coordinates. Taking the $j$th term from the $j$th nested subsequence produces one subsequence that converges in every fixed coordinate.

If $I_1\supseteq I_2\supseteq \cdots$ are nonempty closed bounded intervals, then their intersection is nonempty. If their lengths tend to zero, the intersection consists of exactly one point.

If a bounded real sequence has the property that every convergent subsequence converges to the same number $L$, then the full sequence converges to $L$. Otherwise, a subsequence stays at least some fixed distance from $L$; Bolzano--Weierstrass gives it a convergent subsubsequence, contradicting the assumed uniqueness.