Supposem $X1,X2,...$ are i.i.d. and $X1$ takes each value $j$ with probability $p_j$. Let $D_n$ be the number of distinct values $j$ among $X1,X2,...X_n$, that is, $D_n=|{X1,X2,...,X_n}|
(i) Show that $D \rightarrow \infty$ a.s.
(ii) show that $ED_n/n \rightarrow 0$.
My solution: Let Define the events $A_j=\{j \in D_n \ \text{for some} n\}$. So, $P(A_j^c)=P\{j \notin D_n \ \text{for all} \ n \}=lim (1-p_j)^n=0$. Hence, $P(A_j)=1$.
We have
$P(D_n \rightarrow \infty)=P(A_j \ \text{i.o.})=P(limsup A_j)\geq limsup P(A_j)=1$
I am not sure my solution is correct.
For (ii), I wanted to calculate the expectation but it seems to be hard to calculate it.