1. simple uniform convergence q

Determine directly from the definition whether

$f_{n}(x) = \frac{x}{1 + nx}$ converges uniformly to $0$ on $[0,\infty).$

2. If $x=0,f_n(0)=\left.\frac{x}{1+nx}\right|_{x=0}=0,\fo rall n\in\mathbb{N}.$

If $x\not=0\Rightarrow 0\leq \frac{x}{1+nx}=\frac{1}{\frac{1}{x}+n}\leq \frac{1}{n}.$

Then $\sup_{x\in[0,\infty)}\left|\frac{x}{1+nx}-0\right|\leq \frac{1}{n}<\epsilon.$