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.
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Hint: let $f(x)=xe^{-nx}$. For which $x$ is that a maximum? – lulu May 07 '17 at 12:27
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This is equivalent to $$ e^{nx}\ge nx $$ or $$ e^t\ge t $$ Consider $f(t)=e^t-t$, compute its minimum and conclude.
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