Suppose (a_{n}) is bounded, and that every convergent subsequence of (a_{n}) has limit L. Prove that $\displaystyle \lim_{n\rightarrow\infty}a_{n} = L$

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- Mar 17th 2013, 12:48 PMKanwar245Bounded Sequence
Suppose (a

_{n}) is bounded, and that every convergent subsequence of (a_{n}) has limit L. Prove that $\displaystyle \lim_{n\rightarrow\infty}a_{n} = L$ - Mar 17th 2013, 01:15 PMPlatoRe: Bounded Sequence
- Mar 17th 2013, 01:23 PMKanwar245Re: Bounded Sequence
Okay so from Bolzano-Wiererstrass since a_n is bounded, there exists atleast one convergent subsequence. Let that subsequence be a_n itself and we're done?

- Mar 17th 2013, 01:39 PMPlatoRe: Bounded Sequence
Suppose the sequence does not converge to $\displaystyle L$.

What does that mean?

If it does converge to $\displaystyle L$ then if $\displaystyle \varepsilon > 0$ and $\displaystyle (L - \varepsilon ,L + \varepsilon )$ contains almost of the terms. So what is the negation of that? - Mar 17th 2013, 01:51 PMHallsofIvyRe: Bounded Sequence
- Mar 17th 2013, 01:54 PMKanwar245Re: Bounded Sequence