Prove that $\limsup(a+b) \leq \limsup(a) + \limsup(b)$ and $\liminf(a+b) \geq \liminf(a) + \liminf(b)$
Prove $\limsup\limits_{n \to \infty} (a_n+b_n) \le \limsup\limits_{n \to \infty} a_n + \limsup\limits_{n \to \infty} b_n$