Putnam 1963 B5

Let $(a_n )$ be a sequence of real numbers satisfying the inequalities $$0 \leq a_k \leq 100a_n \; \text{for} \; n \leq k \leq 2n \; \text{and} \; n=1,2,\ldots,$$ and such that the series $$\sum_{n=0}^{\infty} a_n$$ converges. Prove that $$\lim_{n\to \infty} n a_n = 0.$$