Computing the limit of function containing a power series.

Prove that if the sequence $a_{n}$ of real numbers converges to a finite limit;

$$\begin{align} \lim_{n \rightarrow \infty} a_{n} = g, \end{align}$$

then
$$\begin{align} \lim_{x \to \infty} \left({\rm e}^{-x}\sum_{n = 0}^{\infty}a_{n}\,{x^{n} \over n!}\right) = g. \end{align}$$
The initial observation is the power series of $e^{x}$ is given by
$$\begin{align} e^{x} = \sum_{n = 0}^{\infty} \frac{x^{n}}{n!}. \end{align}$$
I want to use summation by parts somehow while using some sort of telescoping technique. Is this the right technique? How do I get started with this?