Problem $3[4+4=8$ marks$]$

Are the following two statements true or false? If a statement is true, provide a formal

proof based on the definitions of the order notations. If a statement is false, provide a

counterexample and explain it$.$ Functions f and g are positive functions.

a$)$ If h$($n$)=(1)$ and f$($n$)=($g$($n$),$ then f$($n$)=($g$($n$)\backslash$times h$($n$)).$ We assume that

h$($n$)>0$ for all n$.$

b$)$ If f$1($n$)=$ O$($g$1($n$)),$ f$2($n$)=($g$2($n$))$ then f$1($n$)$ f$2($n$)=$ O$($g$1($n$)$ g$2($n$)).$ We

assume that f$1($n$)$ f$2($n$)>=0$ and g$1($n$)$ g$2($n$)>=0$ for all n$.$