Bolzano–Weierstrass theorem

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