Let {x(adsbygoogle = window.adsbygoogle || []).push({}); _{n}}_{n = 1}^{[itex]\infty[/itex]}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}^{[itex]\infty[/itex]}converges to L.

My proof is as follows.

Suppose that {x_{n}}_{n = 1}^{[itex]\infty[/itex]}does not converge to L; this implies that there is a subsequence |{x_{nj}}-L|[itex]\geq[/itex][itex]\epsilon[/itex]. 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?

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# Homework Help: Application of Bolanzo-Weierstrauss

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