Bolzano–Weierstrass theorem

Every bounded sequence in Euclidean space has a convergent subsequence. It characterizes sequential compactness of closed, bounded subsets of finite-dimensional Euclidean space.

Bolzano–Weierstrass theorem for real numbers: Every bounded real sequence has a convergent subsequence. Repeatedly bisecting a bounded interval gives the theorem's one-dimensional proof.
Bounded set: A set whose elements all lie within some finite distance of a fixed point. Boundedness prevents the sequence from escaping every finite region.
Extreme value theorem: A continuous real-valued function on a compact space attains both a maximum and a minimum. The theorem helps prove extrema exist by extracting limits from optimizing sequences.
Unit ball in an infinite-dimensional normed space: The set of vectors with norm at most one in an infinite-dimensional normed space. It is bounded but need not be compact, so sequences can lack norm-convergent subsequences.
Nested interval theorem: A nested sequence of closed intervals with lengths tending to zero has exactly one common point. The interval-bisection proof traps a subsequence around this unique point.
Convergent sequence: A sequence whose terms approach a fixed limit as their indices increase. The conclusion identifies a subsequence that approaches one point.
Weierstrass theorem: A continuous function on a compact space is bounded and attains its bounds. Sequential compactness yields convergent subsequences used in its extremum arguments.
Compact space: A topological space in which every open cover has a finite subcover. Compactness guarantees convergent subsequences in metric spaces, beyond Euclidean boundedness alone.
Cauchy sequence: A sequence whose terms become arbitrarily close to one another beyond some index. In Euclidean space, the convergent subsequence is Cauchy, linking compactness to completeness.
Subsequence: A sequence formed by selecting terms from another sequence in strictly increasing index order. The theorem guarantees convergence after discarding some terms while preserving their order.

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