Is $\frac{nx}{n^{2}x^{2}+1}$ Cauchy in uniform norm (I guess it is not Cauchy but am not sure about my proof)?

Question

The sequence $f_{n}(x)=\frac{nx}{n^{2}x^{2}+1}$, for $x\in[0,1]$, pointwise converges function $f(x)=0$.

Is $\left\{f_{n}(x)\right\}$ Cauchy with the norm $||\cdots||_{\infty}$?

Answer 1

Hint:

$\displaystyle \sup_{x\in [0,1]}|f_n(x)-f(x)|=\sup_{x\in [0,1]} f_n(x)\ge f_n\left(\frac{1}{n}\right)=1/2\not \rightarrow 0$