Q:

Convergence uniform when you can't find sup

I have a question that is bugging me and I tried hard to solve it.
Does the functions $f_n(x)=\frac {nx}{1+e^{nx}}$ and $g_n(x)=\frac {\frac nx}{1+e^{\frac nx}}$ pointwise convergence?
For $f_n(x)$  we can prove that for all $x \ge 0$ the function pointwise convergence.
It is easy to prove that as $n$ approaches $\infty$ the function approaches $f(x)=0$
To prove that a function uniformly converges, we can prove that $b_n = |f_n(x)-f(x)| < \epsilon$ 
Or in other words, I must prove that supreme of $b_n$ tends to zero.
The problem is that I can not find a supreme of the function $f_n(x)$ , apparently the supreme does not exist. How would I go to find out if the function uniformly converges?

A:

You don't actually have to find an exact value for the supremum to show this sequence fails to converge uniformly.   
Notice that at $x = 1/n$ we have $f_n(1/n) = 1/(1+e) > 0.$ 
Thus, $\sup_{x \in [0,\infty)}f_n(x) \geqslant 1/(1+e)$ and consequently $\lim_{n \to \infty} \sup_{x \in [0,\infty)}f_n(x) \neq 0$.
The same argument applies for $g_n$ using $x_n = n$. The non-uniform convergence for $f_n$ ($g_n$) is associated with $x \to 0$ ($x \to \infty)$.

