I have series $$\sum_{n=0}^{\infty} 1/(1 + n^2x^2) $$ for $x \in (0,1]$. I think this will not converge uniformly if we pick $x = 1/n$ since $x \in (0,1]$ and we can see that this will contradict the original definition for uniform convergence because we can see for interval $(0,1]$ the point wise limit is $0$ so picking $x = 1/n$ we and choosing $\varepsilon = 1/2$
$|\sum 1/(1 + 1)| $ = $\sum 1/2$ > 1/2 so it will not uniformly converge. What is wrong with this argument?