Let $a_k$ be an integer valued sequence, $a_k \in \mathbb{N}^+$ and let $b_k = \#\{i: a_i=1,\; i \leq k\}$ and assume that $b_k=o(k)$ (little o notation).
How to prove that there exists a constant $c_+ < \infty$ depending only on $a_k$ s.t. $\forall n\geq 1$: $$ \sum_{i=0}^n \prod_{j=0}^ia_j\leq c_+\prod_{i=0}^na_i$$
Thank you for your help!