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In mathematics, specifically in real analysis, the Bolzano–Weierstrass theorem, named after Bernard Bolzano and Karl Weierstrass, is a fundamental result about convergence in a finite-dimensional Euclidean space R''n''. The theorem states that each bounded sequence in R''n'' has a convergent subsequence.〔Bartle and Sherbert 2000, p. 78 (for R).〕 An equivalent formulation is that a subset of R''n'' is sequentially compact if and only if it is closed and bounded.〔Fitzpatrick 2006, p. 52 (for R), p. 300 (for R''n'').〕 The theorem is sometimes called the sequential compactness theorem.〔Fitzpatrick 2006, p. xiv.〕 == Proof == First we prove the theorem when ''n'' = 1, in which case the ordering on R can be put to good use. Indeed, we have the following result. Lemma: Every sequence in R has a monotone subsequence. Proof: Let us call a positive integer ''n'' a "peak of the sequence" if ''m'' > ''n'' implies ''i.e.'', if ''x''''n'' is greater than every subsequent term in the sequence. Suppose first that the sequence has infinitely many peaks, ''n''1 < ''n''2 < ''n''3 < … < ''n''''j'' < …. Then the subsequence corresponding to these peaks is monotonically decreasing, and we are done. So suppose now that there are only finitely many peaks, let ''N'' be the last peak and . Then ''n''1 is not a peak, since , which implies the existence of an with Again, is not a peak, hence there is with Repeating this process leads to an infinite non-decreasing subsequence , as desired.〔Bartle and Sherbert 2000, pp. 78-79.〕 Now suppose we have a bounded sequence in R; by the Lemma there exists a monotone subsequence, necessarily bounded. It follows from the monotone convergence theorem that this subsequence must converge. Finally, the general case can be easily reduced to the case of ''n'' = 1 as follows: given a bounded sequence in R''n'', the sequence of first coordinates is a bounded real sequence, hence has a convergent subsequence. We can then extract a subsubsequence on which the second coordinates converge, and so on, until in the end we have passed from the original sequence to a subsequence ''n'' times — which is still a subsequence of the original sequence — on which each coordinate sequence converges, hence the subsequence itself is convergent. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Bolzano–Weierstrass theorem」の詳細全文を読む スポンサード リンク
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