Let $0 \leqslant \alpha<1$ and $A>0$. If we have an infinite sequence of integers $m_{n}$ with $1 \leqslant m_{n} \leqslant A n^{\alpha}$, show that

$$\sum_{n=1}^{\infty} \frac{m_{n}}{n !}$$

is irrational.

Does the result remain true if the $m_{n}$ are not restricted to integer values? Justify your answer.