Proving that $f_n(x)=\frac{nx}{2+n+x}$ converges uniformly on $0 \le x \le 1$
Now I know I have to use the infinity metric, but I can't understand the solution given for this question.
The next step is $f_n(x)-x=-\frac{2x+x^2}{2+x+n}$ and that is then used to prove, yet I'm not sure as to why that route is taken for this question. I'd appreciate any hints or explanations.
One has, as $n \to \infty$, $$ \sup_{x \in [0,1]}\left|f_n(x)-x\right|=\sup_{x \in [0,1]}\left|\frac{2x+x^2}{2+x+n}\right|\leq \frac{3}{2+n} \to 0. $$