Showing that a sequence of function does not converge uniformly [duplicate]

let $f_n(x)$ = $\frac{nx}{1+nx^2}$ for $x\ge 0$,

(a) show that $\{f_n\}$ converges uniformly on $[M,\infty]$ for any $M>0$

(b) show that $\{f_n\}$ does not converge uniformly on $[0,\infty)$.

I tried solving this problem by using Weierstrass $M_n$ test for sequence of functions:

$$f_n(x) \to f(x)=\frac{1}{x}.$$

Now, I am stuck in finding

$$M_n=\sup\{\left\lvert f_n(x)-f(x)\right\rvert\},$$ where the supremum is taken over $[M,\infty)$ for (a) or $[0,\infty)$ for (b).

Also, is it true in general that if $M_n$ does not tend to $0$, then $\{f_n\}$ does not uniformly converge to $f$?

Hint 1: For $|x|\ge M$, $$ \begin{align} \left|\frac{nx}{1+nx^2}-\frac1x\right| &=\frac1{x(1+nx^2)}\\ &\le\frac1{nM^3} \end{align} $$ Hint 2: The Uniform Limit Theorem.