US-TST 2011 Problem 3

Let $p$ be a prime. We say that a sequence of integers $\{z_n\}_{n=0}^\infty$ is a $p$ -pod if for each $e \geq 0$ , there is an $N \geq 0$ such that whenever $m \geq N$ , $p^e$ divides the sum $$\sum_{k=0}^m (-1)^k {m \choose k} z_k.$$ Prove that if both sequences $\{x_n\}_{n=0}^\infty$ and $\{y_n\}_{n=0}^\infty$ are $p$ -pods, then the sequence $\{x_ny_n\}_{n=0}^\infty$ is a $p$ -pod.