An equality involving limit superior

Question

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.

Answer 1

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 $\displaystyle \limsup (a_n + b_n)$?

Answer 2

Show that

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