SCHWEITZER 2002 Problem 10

Let $X_1, X_2, \ldots$ be independent random variables of the same distribution such that their joint distribution is discrete and is concentrated on infinitely many different values. Let $a_n$ denote the probability that $X_1,\ldots, X_{n+1}$ are all different on the condition that $X_1,\ldots, X_n$ are all different ( $n\ge 1$ ). Show that (a) $a_n$ is strictly decreasing and tends to $0$ as $n\to \infty$ ; and (b) for any sequence $1\le f(1)\le f(2) < \ldots$ of positive integers the joint distribution of $X_1, X_2, \ldots$ can be chosen such that $$\limsup_{n\to\infty}\frac{a_{f(n)}}{a_n}=1$$ holds.