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I'm trying to prove the following:

Let $(a_n)_{n\in\mathbb{N}}$ and $(b_n)_{n\in\mathbb{N}}$ be two sequences such that $(a_n)_{n\in\mathbb{N}}$ converges and $(b_n)_{n\in\mathbb{N}}$ is bounded. If $a=\lim_{n\to\infty} a_n$, prove that $$\limsup_{n\to\infty} (a_n+b_n) = a +\limsup_{n\to\infty} b_n$$

It's easy to show that $\limsup_{n\to\infty} (a_n+b_n) \leq a +\limsup_{n\to\infty} b_n$ since the limit superior is subadditive, but I'm at a loss on how to prove the other inequality.

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Since $\{b_n\}$ is a bounded sequence, $\lim\sup b_n$ exists as a finite number. On the other hand, there is a subsequence of the sequence $\{b_n\}$, say $\{b_{n_k}\}$, such that $\lim_{k\rightarrow\infty}b_{n_k} = \lim\sup{b_{n_k}}$. Take it from here... –  William Mar 13 '12 at 2:27
@WNY: No matter how trivial an answer might be for you, please consider adding an answer, rather than just commenting. For a few reasons why, please read this meta thread: meta.math.stackexchange.com/questions/1559/…. –  Aryabhata Mar 13 '12 at 2:35
@Aryabhata: I didn't make that comment to point to triviality of the problem (as you said, the level of difficulty depends on one's level of understanding). I made the comment as a hint. In general, if I don't think that the answer is long and elaborate, I prefer to give a hint in the right direction and have the OP do some work (especially in this case, where it seems that the OP is learning the subject and the problem looks like a routine exercise). For instance, your answer below could have very well been a comment :). –  William Mar 13 '12 at 4:34
@WNY: You can always give hints in an answer and have the OP do the work. Once the OP is done, you can edit your answer to add more details etc. I have seen many questions where people just keep giving hints in comments and never bother to provide closure to the question by adding an answer. I hope you did read the meta thread as to why it is good for the site to have answers. As long as you are willing to add an answer later, I suppose it is fine. –  Aryabhata Mar 13 '12 at 5:24
See also this question. –  Martin Sleziak Mar 31 '12 at 7:15

2 Answers 2

up vote 4 down vote accepted

Let $\displaystyle b_{n_k}$ be a subsequence of $\displaystyle b_n$ which converges to $\displaystyle \limsup b_n$.

Show that $\displaystyle a_{n_k} + b_{n_k}$ converges.

How does this limit relate to $\displaystyle \limsup (a_n + b_n)$?

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$a_{n_k} +b_{n_k}\rightarrow a + \limsup b_n$, and since the limit superior is greater or equal than any other limit point, the inequality holds, right? Great answer. –  Fernando Martin Mar 13 '12 at 2:36
@FernandoMartin: Yes you got it! (and Thanks.) –  Aryabhata Mar 13 '12 at 2:38

Show that

$$\forall\epsilon>0:\quad \limsup\, (a_n+b_n)\ge \limsup\, \big((a-\epsilon)+b_n\big)=(a-\epsilon)+\limsup\,b_n $$

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