USAMO 2015 Problem 6

Consider $0<\lambda<1$ , and let $A$ be a multiset of positive integers. Let $A_n=\{a\in A: a\leq n\}$ . Assume that for every $n\in\mathbb{N}$ , the set $A_n$ contains at most $n\lambda$ numbers. Show that there are infinitely many $n\in\mathbb{N}$ for which the sum of the elements in $A_n$ is at most $\frac{n(n+1)}{2}\lambda$ . (A multiset is a set-like collection of elements in which order is ignored, but repetition of elements is allowed and multiplicity of elements is significant. For example, multisets $\{1, 2, 3\}$ and $\{2, 1, 3\}$ are equivalent, but $\{1, 1, 2, 3\}$ and $\{1, 2, 3\}$ differ.)