Homework Help: Application of Bolanzo-Weierstrauss

1. Aug 31, 2011

wrldt

Let {xn}n = 1$\infty$ be a bounded sequence

and

{xnj} be a convergent subsequence, each one converging to L.

Want to show that {xn[}[/itex]n = 1$\infty$ converges to L.

My proof is as follows.

Suppose that {xn}n = 1$\infty$ does not converge to L; this implies that there is a subsequence |{xnj}-L|$\geq$$\epsilon$. However by B-W there exists a subsequence of that subsequence that converges, and it must converge to L. However this is a contradiction.

Is this sufficient?

2. Sep 1, 2011

micromass

What exactly is it that you need to prove??