How to prove these inequalities: $\liminf(a_n + b_n) \leq \liminf(a_n) + \limsup(b_n) \leq \limsup(a_n + b_n)$ [duplicate]
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.