Bolzano–Weierstrass theorem

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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 {\displaystyle \mathbb {R} ^{n}} . The theorem states that each infinite bounded sequence in R n {\displaystyle \mathbb {R} ^{n}} has a convergent subsequence. An equivalent formulation is that a subset of R n {\displaystyle \mathbb {R} ^{n}} is sequentially compact if and only if it is closed and bounded. The theorem is sometimes called the sequential compactness theorem.

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