Limit of the series $a_n = ne^{-n}$ as $n$ goes to infinity

Question

How do I find $\lim_{n\rightarrow\infty}a_n = ne^{-n}$?

This is a question for real analysis, so I can not use L'Hôpital's.

I have this similar limit here. But there is no rigorous proof. I know it is equal to zero. But I can not proof it using the definition of limit nor limit properties.

Answer 1

Note that $e^{-1}$ is just a constant, and as $n$ approaches infinity, just the $n$ term matters, so the limit tends to infinity. I think it won't be too difficult to prove using delta-epsilon.