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Bolzano-Weierstrass theorem (Theorem)

Given any bounded sequence $ (a_n)$ of real numbers, there exists a convergent subsequence $ (a_{n_j})$.

More generally, any sequence $ (a_n)$ in a compact subset of a metric space has a convergent subsequence.



"Bolzano-Weierstrass theorem" is owned by yark. [ full author list (2) | owner history (1) ]
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See Also: convergent sequence, sequentially compact, alternate statement of Bolzano-Weierstrass theorem

Other names:  Bolzano-Weierstraß theorem

Attachments:
proof of Bolzano-Weierstrass Theorem (Proof) by akrowne
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Cross-references: metric space, compact subset, subsequence, convergent, real numbers, sequence, bounded
There are 4 references to this entry.

This is version 7 of Bolzano-Weierstrass theorem, born on 2002-02-18, modified 2007-07-01.
Object id is 2125, canonical name is BolzanoWeierstrassTheorem.
Accessed 13277 times total.

Classification:
AMS MSC: 40A05 (Sequences, series, summability :: Convergence and divergence of infinite limiting processes :: Convergence and divergence of series and sequences)
 26A06 (Real functions :: Functions of one variable :: One-variable calculus)
Pending Errata and Addenda
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