epsilon -N proof of a limit help

[shelly89]

Please can someone explain the proof of this one for me, I just dont get it..

I get
\(\displaystyle | n*e^{-n}-0 | < \epsilon \)

 

[pka]

Please can someone explain the proof of this one for me, I just dont get it..
\(\displaystyle | n*e^{-n}-0 | < \epsilon \)

Note that \(\displaystyle (\forall n)\left[\dfrac{n}{e^n}>0\right] \) so no need for absolute value; also \(\displaystyle (\forall n)\left[e^n>n^2\right] \)

Thus \(\displaystyle (\forall n)\left[\dfrac{n}{e^n}<\dfrac{1}{n}\right] \).

Again by the the Archimedean_property we make \(\displaystyle \dfrac{1}{n}<\epsilon \).