Part $1$: Understanding ALOHA $(10$ Marks$)$

A$)$ You studied in class that the efficiency of Slotted Aloha, with N users and a transmit probability p$,$ is given by:

Efficiency$($slotted Aloha$)=$ Np$(1-$p$)$N$-1$

$$ i$.$ Prove that the probability that maximizes the efficiency of Slotted Aloha is p$*=1/$N

$$ ii$.$ Show that when N tends to infinity, the maximum efficiency of Slotted$-$Aloha reaches the value of $1/$e $=0\mathrm{.}37.($Hint: you can use the fact that the limit when N goes to infinity of $(1-1/$N$)$N $=1/$e$).$