Proving if $\frac{nx}{e^{nx}}$ converges uniformly

Question

I have this expression $$ \frac{nx}{e^{nx}} \qquad x\geq 0 $$ and I had to test if it's pointwise and uniformly convergent(separately). Pointwise convergence could be easily proved by just evaluating the limit below $$ \lim_{n\to\infty}\frac{nx}{e^{nx}}\to 0 $$ To prove uniform convergence I had to evaluate this limit and prove that its equal to $0$ $$ \lim_{n\to\infty}\left|\left|\frac{nx}{e^{nx}} - 0\right|\right| $$ I'm a little confused by how im supposed to calculate the above, is the supremum with respect to $n$ or $x$? Sorry, I'm new to this.

Answer 1

It does not uniformly converge to $0$, as $$\sup_{x \geq 0} \frac{nx}{e^{nx}} = \frac{n \times \frac{1}{n}}{e^{n \frac{1}{n}}} = \frac{1}{e}$$ which doesn't tend to $0$ as $n \to \infty$.