Q:

Sum of limit inferior

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.

A:

So, I will do the analogous problem involving $\limsup$, leaving $\liminf$ up to you! :) I think that will be more beneficial to you anyway. (Study this problem first. Understand it. Then go back to your problem.)
Since both $(s_n)$ and $(t_n)$ are bounded, $(s_n + t_n)$ is bounded. Set $s = \limsup s_n$ and $t = \limsup t_n$, both of which are in $\mathbb{R}$. Let $\varepsilon > 0$. Since $s = \limsup s_n$, there exists an $N_1 \in \mathbb{N}$ such that, for every $n \in \mathbb{N}$, $n \geq N_1$ implies $s_n < s + \varepsilon/2$. Since $t = \limsup t_n$, there exists an $N_2 \in \mathbb{N}$ such that, for every $n \in \mathbb{N}$, $n \geq N_2$ implies $t_n < t + \varepsilon/2$. 
Set $N = \max \{ N_1, N_2 \}$. Let $n \in \mathbb{N}$, and suppose $n \geq N$. Then $n \geq N_1$ and $n \geq N_2$, which implies
$$
 s_n + t_n < \biggl( s + \dfrac{\varepsilon}{2} \biggr) + \biggl( t + \dfrac{\varepsilon}{2} \biggr) = s + t + \varepsilon.
$$
Since $\varepsilon$ was arbitrary, we have shown that there are (at most) finitely many $n \in \mathbb{N}$ for which $s_n + t_n \geq s + t + \varepsilon$; no subsequence of $(s_n + t_n)$ can converge to a number greater than $s + t$. That is, $\limsup {(s_n + t_n)} \leq s + t$.

