Prove that in any set $A_\epsilon=(-\infty,-\epsilon]\cup[\epsilon, \infty)\subset \mathbb{R}$, where $\epsilon>0$, the convergence of $\forall x \in \mathbb{R}\:\: \lim_{n\to\infty} f_n(x)\stackrel{pw}{=}f(x) = \begin{cases}\begin{align} 1 \quad&\text{ if }\; x\neq0, \\ 0 \quad&\text{ if }\; x=0 \\ \end{align}\end{cases}$ is uniform.
Where $f_n(x)=\frac{nx^2}{nx^2+1}, x\in\mathbb{R}$
The first part of the question is to prove that the equation I listed is not uniform and I did that using Lebesgue's Monotone Convergence Theorem. It seems intuitive that when you don't include 0 as is the case here (because epsilon is arbitrarily small but not 0) that it should be convergent but I don't know how to prove. The section that we are in talks a fair bit about Lesbesgue and also mentions the Weierstrab M-Test (though I don't think that applies) and is about commuting limits with integrals and derivatives.
Any help would be appreciated.