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Every bounded sequence in Euclidean space has a convergent subsequence
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 Euclidean space is a real inner product space with .
A bounded set in a metric space is a subset for which there exist and such that , where is the open ball of radius around .
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.
A convergent sequence in a topological space converges to a point if for every neighborhood of , there exists such that for all .
In a metric space , this is equivalent to . In a Hausdorff space, limits of convergent sequences are unique (see uniqueness of limits).
Heine–Borel theorem: In with its usual metric (and induced topology), a set is compact if and only if it is both closed and bounded.
This specializes general implications like compactness implies boundedness and compactness implies closedness into a complete characterization in Euclidean settings, and it is frequently paired with Bolzano–Weierstrass and sequential criteria such as sequential compactness.
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.