Proving that $\liminf (x_n)+\liminf (y_n)\le \liminf (x_n+y_n)$ I know that $\limsup (x_n)+\limsup (y_n)\ge \limsup (x_n+y_n)$, but I'm not sure how to show that $\liminf (x_n)+\liminf (y_n)\le \liminf (x_n+y_n)$. Please help me.
Start with the inequality for $\limsup$ that you know, and use the hint of RRL.
\begin{align}
\limsup (-x_n) + \limsup (-y_n) &amp;\ge \limsup (-x_n-y_n)\\
-\liminf x_n - \liminf y_n &amp;\ge -\liminf (x_n+y_n)\\
\liminf x_n + \liminf y_n &amp;\le \liminf (x_n+y_n)
\end{align}
Fix $n$ for the moment and let $l \ge n$. Since $\inf_{ k \ge n} x_k \le x_l$ and $\inf_{ k \ge n} y_k \le y_l$, we see that
$\inf_{ k \ge n} x_k + \inf_{ k \ge n} y_k \le x_l + y_l$. Hence
$\inf_{ k \ge n} x_k + \inf_{ k \ge n} y_k \le \inf_{ k \ge n} ( x_k + y_k)$.

Since these are non decreasing sequences, we have
$\inf_{ k \ge n} x_k + \inf_{ k \ge n} y_k \le \liminf_{ k } ( x_k + y_k)$ and
hence $\liminf_{ k } x_k + \liminf_{ k } y_k \le \liminf_{ k } ( x_k + y_k)$.