Let a_{n}and b_{n}be two convergent sequences. Prove: if for every evenn: a_{n}<=b_{n, }and for every oddn: a_{n}>=b_{n,}then lim a_{n}= lim b_{n}.

Any ideas? I'm looking for a formal proof.

Thanks!

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- July 9th 2012, 04:24 AMloui1410Limit of two sequences
Let a

_{n}and b_{n}be two convergent sequences. Prove: if for every even*n*: a_{n}<=b_{n, }and for every odd*n*: a_{n}>=b_{n,}then lim a_{n}= lim b_{n}.

Any ideas? I'm looking for a formal proof.

Thanks! - July 9th 2012, 05:14 AMemakarovRe: Limit of two sequences
This can be proved using the following facts.

(1) The limit of a subsequence of a converging sequence equals the limit of the sequence.

(2) If a_{n}and b_{n}are converging sequences and a_{n}≤ b_{n}for all n, then lim a_{n}≤ lim b_{n}. - July 10th 2012, 07:56 AMloui1410Re: Limit of two sequences
The said exercise is presented in the book before defining subsequences and fact (1), so I'm not supposed to use it. That's why I was looking for a more basic proof,

*almost*only by the definition of limit of sequence. However, I can use the following rules, in case they're helpful:

"If a sequence converges, then its limit is unique."

"Every convergent sequence is bounded." - July 10th 2012, 08:17 AMemakarovRe: Limit of two sequences
Let A = lim aₙ and B = lim bₙ. For a given ε > 0, choose N such that for all n > N we have |aₙ - A| < ε and |bₙ - B| < ε. Pick any even n > N; then inequalities in the previous sentence imply that A < aₙ + ε and bₙ < B + ε. Therefore, A < aₙ + ε ≤ bₙ + ε < B + 2ε. Thus, for the given ε we showed that A < B + 2ε. Since this holds for any positive ε, it must be that A ≤ B. Similarly, you can show that A ≥ B.

- July 10th 2012, 08:33 AMPlatoRe: Limit of two sequences
- July 10th 2012, 08:49 AMloui1410Re: Limit of two sequences
emakarov, thank you - that's a great solution, I wouldn't have thought about it.

Plato's solution, however, seems more natural to me as it's similar to some other examples I've already seen.

Thank you both.