$\limsup_{n\to \infty} a_n+b_n\le \limsup_{n\to \infty} a_n+\limsup_{n\to \infty} b_n$ for real sequence.
Prove $\limsup\limits_{n \to \infty} (a_n+b_n) \le \limsup\limits_{n \to \infty} a_n + \limsup\limits_{n \to \infty} b_n$