How to show that series $\sum\frac{x}{1+n^2x}$is uniform convergent in $(t,1]$ where $t>0$ but not uniform convergent in $[0,1]$?

Question

How to show that series $$\sum\frac{x}{1+n^2x}$$is uniform convergent in $(t,1]$ where $t>0$ but not uniform convergent in $[0,1]$? Please help, it is given in many books, but I find it uniform convergent throughout $[0,1]$. I used weierstrass M test to show that mod$\frac{x}{1+n^2x}$ < $\frac{x}{n^2x}= \frac{1}{n^2}$. Now since $\sum\frac{1}{n^2}$ is convergent hence it is uniform convergent in $[0,1]$