Let $\displaystyle (a_n) $n from 1 to infinity and $\displaystyle (b_n) $n from 1 to infinity be bounded sequences of real numbers. Prove that $\displaystyle sup_{n\inN}$$\displaystyle (a_n+b_n)$ <= $\displaystyle sup_{n\inN}$ $\displaystyle a_n$ $\displaystyle + sup_{n\inN}$ $\displaystyle b_n$.

n$\displaystyle \in$ N

Conclude that $\displaystyle limsup_{n--> infinity} $$\displaystyle (a_n +b_n)$ <= $\displaystyle limsup_{n--> infinity} $$\displaystyle a_n$ + $\displaystyle limsup_{n--> infinity}$ $\displaystyle b_n$. Formulate analogus results for inf and liminf (no need to prove those).

The example i could think of is $\displaystyle a_n = (-1)^n$ and $\displaystyle b_n = (-1)^{n+1}$. 0<2. [tex] How to prove it in general?