Bounded Sequence

• Mar 17th 2013, 12:48 PM
Kanwar245
Bounded Sequence
Suppose (an) is bounded, and that every convergent subsequence of (an) has limit L. Prove that $\displaystyle \lim_{n\rightarrow\infty}a_{n} = L$
• Mar 17th 2013, 01:15 PM
Plato
Re: Bounded Sequence
Quote:

Originally Posted by Kanwar245
Suppose (an) is bounded, and that every convergent subsequence of (an) has limit L. Prove that $\displaystyle \lim_{n\rightarrow\infty}a_{n} = L$

Well any sequence is a subsequence of itself.
• Mar 17th 2013, 01:23 PM
Kanwar245
Re: 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 PM
Plato
Re: Bounded Sequence
Quote:

Originally Posted by Kanwar245
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?

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 PM
HallsofIvy
Re: Bounded Sequence
Quote:

Originally Posted by Kanwar245
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?

No, you can say that there exist at least one convergent subsequence but you [b]cannot[b] assert that it happens to be [itex]\{a_n\}[/itself]. You still need to prove that the sequence is itself convergent. Do, rather, what Plato suggested.
• Mar 17th 2013, 01:54 PM
Kanwar245
Re: Bounded Sequence
Quote:

Originally Posted by HallsofIvy
No, you can say that there exist at least one convergent subsequence but you [b]cannot[b] assert that it happens to be [itex]\{a_n\}[/itself]. You still need to prove that the sequence is itself convergent. Do, rather, what Plato suggested.

okay