I'm seeking to prove the following statement:
$$\lim \inf s_{n} + \lim \inf t_{n} \le \lim \inf (s_{n}+t_{n})$$provided that $s_{n}$ and $t_{n}$ are bounded.
My solution so far:
Given that $s_{n}$ and $t_{n}$ are bounded, $\lim \inf s_{n}$ and $\lim \inf t_{n}$ both exist and are real numbers. Call them $s$ and $t$ respectively. Now since $\lim \inf s_{n} = s$, given some $\epsilon > 0$, the inequality $s_{n} > s + \epsilon$ fails for a finite amount of $n$'s. These $n$'s then have the property that $s_{n} + t_{n} \le s + t_{n} + \epsilon$...
So here I was able to find an inequality relating $s_{n}$ and $t_{n}$ but I seem to have it in the reverse order. Nor do I understand how to preserve the inequality if I argue about applying $\lim \inf$ to $s_{n}$ and $t_{n}$ in the inequality.
Any help would be appreciated.