Evaluate :
$$ \lim_{n\to\infty}\left(\dfrac{1}{e^{n}}\displaystyle \sum_{r=0}^{n} \dfrac{n^r}{r!}\right) $$
Numerical calculation suggests that the limit should be $\dfrac{1}{2}$. I tried using Squeeze Theorem and Stolz - Cesaro Theorem, but to no avail.
An elementary solution is appreciated.
Thanks in advance.