To prove that $f_n= nxe^{-nx}$ is not uniformly convergent on $(0,\infty)$
I came up with a proof, But need to check whether that is correct or not...
Proof:
It is easy to see that $f_n$ converges pointwise to $f$ where $f(x)=0, \forall x \in (0, \infty)$.
Suppose it is uniformly convergent, then given $\epsilon>0, \exists N \in \mathbb{N},s.t. \forall n\ge N, \forall x \in (0,\infty) , |nxe^{-nx}|<\epsilon$
In Particular $\forall x \in (0,\infty),|Nxe^{-Nx}|<\epsilon$, but let $x=\frac{1}{N}$, which implies $|e^{-1}|< \epsilon$, so If I choose my $\epsilon\leq\frac{1}{e}$, there comes a problem and hence I can take $\epsilon=\frac{1}{e}$ and complete the proof.
Is this argument looks clean and neat? If not how can I correct it