consider $f_n(x)$=$\frac{nx}{1+nx^2}$ suppose we talk about it's uniform convergence on [0,$\infty$)
now, $f_n(x)\to f(x)$, where $f(x)=\begin{cases} \frac{1}{x}, & \text{if } x \neq 0 \\[2ex] 0, & \text{if } x=0. \end{cases}$
Now, I am sure that $f_n(x)$ is not uniformly convergent to $f(x)$ on $[0,\infty)$ by using uniform limit theorem. But, instead I want to show non uniform convergence in $[0,\infty)$ using Weierstrass test only. Now, I am stuck on what is $M_n =\sup\{\left\lvert f_n(x)-f(x)\right\rvert\}$, where supremum is taken over $[0,\infty)$, please give answer in detail.