For integer $n \geq 1$, define
$$a_n = \sum_{k=0}^{\infty} \frac{k^n}{k!}, \quad b_n = \sum_{k=0}^{\infty} (-1)^k \frac{k^n}{k!}.$$
Prove that $a_n b_n$ is an integer.

Let $a_1\le a_2\le \cdots \le a_n$ be integers. Prove that

is an integer.