Question asks to show that series $\sum\frac{x}{1+n^2x}$ is uniformly convergent on $[\alpha,1]$ for any $\alpha > 0$ but is not uniformly convergent on $[0,1]$
I could show that the above series is uniformly convergent on $[\alpha,1], \alpha > 0$ by using inequality $\frac{x}{1+n^2x} < \frac{x}{1+n^2\alpha} = u_{n}$, which is convergent by Raabe's test and hence the series is uniformly convergent on $[\alpha,1]$ by Weierstrass M-test
But i am clueless how to show it non uniformly convergent on $[0,1]$