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 Rⁿ. The theorem states that each bounded sequence in Rⁿ has a convergent subsequence. An equivalent formulation is that a subset of Rⁿ is sequentially compact if and only if it is closed and bounded. The theorem is sometimes called the sequential compactness theorem.

The Bolzano–Weierstrass theorem is named after mathematicians Bernard Bolzano and Karl Weierstrass. It was actually first proved by Bolzano in 1817 as a lemma in the proof of the intermediate value theorem. Some fifty years later the result was identified as significant in its own right, and proved again by Weierstrass. It has since become an essential theorem of analysis.

First we prove the theorem when ${\displaystyle n=1}$, in which case the ordering on ${\displaystyle \mathbb {R} }$ can be put to good use. Indeed, we have the following result.

...
Wikipedia