This question already has an answer here:
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.