I need to test whether the sequence of functions $f_n$ given by $f_{n}(x)= n^2(e^{x/n} -1 -x/n)$ converges uniformly over $\mathbb{R}$ or not.
I know this sequence converges pointwise to $x^2/2$.
Now I consider $M_n = \text{sup}_ {x \in \mathbb{R}}\left|n^2(e^{x/n} -1 -x/n)- x^2/2\right|$
$M_n \ge \left|n^2(e^{n/n} -1 -n/n)- n^2/2\right|$
as $n \to \infty$ ,$M_n \to \infty$ , so this is not uniformly convergent.
Is this proof correct and rigorous enough ?
Thank you.