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 . The theorem states that each infinite bounded sequence in has a convergent subsequence. An equivalent formulation is that a subset of 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 for (set of all real numbers), in which case the ordering on can be put to good use. Indeed, we have the following result: Lemma: Every infinite sequence in has a monotone subsequence. Proof: Let us call a positive integer-valued index of a sequence a "peak" of the sequence when for every . Suppose first that the sequence has infinitely many peaks, which means there is a subsequence with the following indices and the following terms . So, the infinite sequence in has a monotone subsequence, which is . But suppose now that there are only finitely many peaks, let be the final peak and let the first index of a new subsequence be set to . Then is not a peak, since comes after the final peak, which implies the existence of with and . Again, comes after the final peak, hence there is an where with . Repeating this process leads to an infinite non-decreasing subsequence , thereby proving that every infinite sequence in has a monotone subsequence. Now suppose one has a bounded sequence in ; by the lemma proven above there exists a monotone subsequence, likewise also bounded. It follows from the monotone convergence theorem that this subsequence converges.