I am given $n$ balls. For $n$ times, I pick one of them with uniform probability and put it back after picking it.

Let $U$ be the number of balls I have never picked, so $U\in \{0,\ldots,n-1\}$. We are interested in the expected value $E_n:= E(U)$.

What is the value of $\lim_{n\to\infty} \frac{E_n}{n}$, if it exists?

(This is a follow-up question to this question.)