Exponentials to the negative power

Question

Could someone please help me in showing how $$xe^{-nx}\le\frac1n$$ I don't totally see why the LHS is bounded by the RHS. Thank you.

Answer 1

This is equivalent to $$ e^{nx}\ge nx $$ or $$ e^t\ge t $$ Consider $f(t)=e^t-t$, compute its minimum and conclude.