Question

Bolzano-Weirstrass Theorem​

Answer 1

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

Bolzano-Weierstrass theorem states that every bounded sequence has a limit point. But, the converse is not true. That is, there are some unbounded sequences which have a limit point.

The Bolzano-Weierstrass Theorem says that no matter how “random” the sequence (xn) may be, as long as it is bounded then some part of it must converge. This is very useful when one has some process which produces a “random” sequence such as what we had in the idea of the alleged proof in Theorem 7.3.

Hope this helps mate!