I am trying to solve this probability problem.
Let $(\epsilon_n)$ be a sequence of nonnegative integer iid random variables. We consider $X_n$ the number of distinct values of the set of $\{\epsilon_1,\ldots, \epsilon_n\}$. Prove that $\frac{X_n}{n} \rightarrow 0$ a.s.
If the variable $\epsilon$ is bounded the solution is trivial. I also observe that $X_{n+1}$ can take only the two values $X_n$ or $X_n+1$ so I tried to compute the probability of $\{X_n +1 = X_{n+1}\}$ but without any assumption on the distribution of $\epsilon$ it seems to be hard. Can any one help me with an idea ?