The Bolzano-Weierstrass Theorem states that every bounded sequence of real numbers has a convergent subsequence. Named for Bernard Bolzano and Karl Weierstrass, it is a foundational compactness result in real analysis and underlies proofs of the Heine-Borel Theorem and the existence of extrema of continuous functions on closed bounded intervals.
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StatementEvery bounded sequence in a finite dimensional Euclidean space has a convergent subsequence. 1 Classification
Statement Form Bolzano-Weierstrass Theorem
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1. Bolzano-Weierstrass Theorem (Wikipedia)
Wikimedia FoundationHistory and significance section
It was actually first proved by Bolzano in 1817 as a lemma in the proof of the intermediate value theorem.
Lead section
The theorem is sometimes called the sequential compactness theorem.
Principal Result Of: Bernard Bolzano, Lead sentence of the article on the result
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 {\displaystyle \mathbb {R} ^{n}} .
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