# $\lim_{n\rightarrow\infty}\sup(a_n+b_n) = \lim_{n\rightarrow\infty}\sup a_n + \lim_{n\rightarrow\infty}\sup b_n$ for all bounded sequences $(b_n)$

Let $(a_n)$ be a bounded sequence. Suppose that for every bounded sequence $(b_n)$ we have $\lim_{n\rightarrow\infty}\sup(a_n+b_n) = \lim_{n\rightarrow\infty}\sup a_n + \lim_{n\rightarrow\infty}\sup b_n$. Prove that $(a_n)$ is convergent.

We can take $b_n=-a_n$ to get $\lim_{n\rightarrow\infty}\sup a_n + \lim_{n\rightarrow\infty}\sup (-a_n) = 0$. For any subsequence of $a_n$ which converges to $L$, the corresponding subsequence of $-a_n$ converges to $-L$. How can we conclude from here?

Hint: $$\lim_{n\to\infty}\sup (-a_n)=-\lim_{n\to\infty}\inf a_n.$$