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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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    See also [this question](http://math.stackexchange.com/questions/33888/limsup-x-ny-n-lim-x-n-limsup-y-n).2012-03-31

2 Answers 2

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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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    @Fer$n$andoMartin: Yes you got it! (and Thanks.)2012-03-13
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Show that

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