Estimation of the expected number of sites visited by i.i.d

Let $$X_1,X_2,...$$ be i.i.d variables with value in $$\mathbb{N}$$ (not necessarily finitely supported). Suppose $$E(X_1) < \infty$$.

Denote :

$$R_n = \textbf{Card}\{X_1,...,X_n\}$$

I must prove that $$E(R_n) = o(\sqrt{n})$$, but do not really know how to proceed. Does anyone have a hint :)? Many thanks !

Denote $$p_k=P(X=k)$$. Then $$E(R_n)=\sum_k P(k\in \{X_1,\ldots,X_n\})\leqslant \sum_k \min(1,np_k)$$. We are given that $$\sum kp_k<\infty$$. Fix $$\varepsilon>0$$. The sum of $$\min(1,np_k)$$ over $$k<\varepsilon \sqrt{n}$$ is of course at most $$1+\varepsilon \sqrt{n}$$. The sum over $$k\geqslant \varepsilon \sqrt{n}$$ is at most $$n\sum_{k\geqslant \varepsilon \sqrt{n}}p_k\leqslant \frac{\sqrt{n}}{\varepsilon}\sum_{k\geqslant \varepsilon \sqrt{n}}kp_k=o(\sqrt{n}).$$ Thus the result.