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A direct corollary form of the Bolzano–Weierstrass theorem
Corollary (Bolzano–Weierstrass, sequence form): If is a bounded sequence in , then there exists a subsequence and a point such that
Connection to parent theorem: This is exactly the Bolzano–Weierstrass theorem, often recorded as a corollary once compactness/sequential compactness is being developed.
A sequence in a metric space is bounded if its range is a bounded set.
A subsequence of a sequence is a sequence of the form , where is a strictly increasing sequence of indices.
Subsequences are a standard way to isolate “partial” limiting behavior of a sequence; they are used in defining limit superior and limit inferior. The monotone subsequence lemma guarantees monotone subsequences under mild hypotheses.
Bolzano–Weierstrass theorem: In Euclidean space with its standard metric, every bounded sequence has a subsequence that is convergent in .
This result is a core compactness phenomenon in Euclidean spaces and is closely tied to the Heine–Borel theorem and to sequential compactness.
A compact set is a subset of a topological space such that every open cover of contains a finite subcover: whenever is a family of open sets with , there exist indices such that
A sequentially compact set is a subset of a topological space such that every sequence in has a subsequence that is a convergent sequence in with limit .
Sequential compactness is phrased purely in terms of sequences, and in many important settings (notably metric spaces) it closely tracks compactness.