Application of Bolanzo-Weierstrauss

[wrldt]

Let {x_n}_{n = 1}^$\infty$ be a bounded sequence

and

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

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

My proof is as follows.

Suppose that {x_n}_{n = 1}^$\infty$ does not converge to L; this implies that there is a subsequence |{x_nj}-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?

 

[micromass]

What exactly is it that you need to prove??