TITLE: How to prove these inequalities: $\liminf(a_n + b_n) \leq \liminf(a_n) + \limsup(b_n) \leq \limsup(a_n + b_n)$
QUESTION [7 upvotes]: The inequalities are:
$$\liminf(a_n + b_n) \leq \liminf(a_n) + \limsup(b_n) \leq \limsup(a_n + b_n)$$

REPLY [9 votes]: Hint: First show that $\limsup$ is subadditive, that is, 
\[
  \limsup(a_n + b_n) \le \limsup a_n + \limsup b_n
\]
for real sequences $(a_n)$, $(b_n)$. From this conclude using $-\limsup(-a_n) = \liminf a_n$ that $\liminf$ is superadditive (inequality $\ge$ in the above). Then you can use all this to prove 
\[ \liminf (a_n + b_n) - \limsup b_n = \liminf (a_n + b_n) + \liminf(-b_n) \le \liminf a_n \]
and the other inequality you need.