Inequalities for limit supremum

Question

We know that for any bounded sequences $(s_n)$ and $(t_n)$ the inequality holds $\limsup (s_n+t_n) \leq \limsup (s_n) + \limsup (t_n)$. I understand how to prove this inequality, but why doesn't this hold for equality; that is, why isn't $\limsup(s_n+t_n) = \limsup(s_n)+\limsup(t_n)$? I would assume that it has something to do with the limits of $\sup\{s_n | n>N\}$ and $\sup\{t_n|n>N\}$.

Answer 1

Here's a simple counterexample:

Let $s_n=(-1)^n$, and let $t_n=(-1)^{n+1}$.

For any $n$, $s_n+t_n=0$. Therefore, $\limsup\limits_{n\to\infty}(s_n+t_n)=0$.

However, $\limsup\limits_{n\to\infty}s_n=1$, and $\limsup\limits_{n\to\infty}t_n=1$.

$$\therefore\ \limsup\limits_{n\to\infty}(s_n+t_n) \ne \limsup\limits_{n\to\infty}s_n + \limsup\limits_{n\to\infty}t_n$$