$(2$ points$)$ Assuming that f$1($n$)$ is O$($g$1($n$))$ and f$2($n$)$ is O$($g$2($n$)),$ prove the following statements, using the

mathematical definition of big$-$O notation:

a$.$ f$1+$ f$2$ is O$($max$($g$1,$ g$2))$

b$.$ k$*$g is O$($g$),$ where k is a positive constant