just looking for a verification on a proof. Thanks in Advance
Let $f_n$ be a sequence of functions such that $f_n=\frac{x^{2n}}{1+x^{2n}}$ defined on $[-2,2]$. Prove or Disprove Uniform Convergence
Proof
Let $x=\sqrt{1+\frac{1}{n}}$ so x $\epsilon (1,2)$ for all $n$. The pointwise limit in this region is 1. But the limit as $n$ goes to infinity of $f_{n}(x)=\frac{e}{1+e}\neq1$ so $f_n$ is not unifromly convergent. $\square$
The proof is globally fine. Maybe it would be better to write $x_n=\sqrt{1+\frac{1}{n}}$ in order to stress on the dependence on $n$, because when you write $f_n(x)$ we could believe that we take $f_n$ of a fixed $x$.