Why isn't $nxe^{-nx}$ uniformly convergent for $x \geq 0$? 
Why isn't $nxe^{-nx}$ uniformly convergent for $x \geq 0$?

The definition of uniform convergence states that $f_n$ is uniformly convergent if
$$\lim_{n\to\infty} ||f_n-f|| = 0.$$
I apply the definition with $f = 0$ and get
$$\lim_{n\to\infty}||\frac{nx}{e^{nx}} - 0 || = 0,$$ 
since $e^{nx}$ grows much faster than $nx$ regardless of $x$ (except when $x = 0$, but if that's the case everything is $0$). 
Where am I wrong?

For any given $x$ it's true that $nxe^{-nx}$ tends to zero as $n\to\infty$. However, the function $f_n$ as a whole, when viewed over all $x\ge0$, will always have a bump somewhere, which keeps the norm $||f_n-f||$ from getting small. The quickest way to see this: $nxe^{-nx}$ is simply the function $xe^{-x}$ but with a compressed $x$-axis. So each $f_n$ has the same maximum value.
