We've been starting to talk about series approximations in my numerical analysis course, and I got to this one series question that has been stumping me. For $n \in \mathbb{N}$ (assume $\mathbb{N}$ doesn't contain a $0$) and $x \in \mathbb{R}$ define $$f_n(x) = \frac{x}{3+nx^2}.$$ I want to show the sequence $\{f_n\}$ converges uniformly to a function $f.$ I have a feeling that I need to first find its bounds for differentiation, because if I can find that the sequence of derivatives converges uniformly on that bound, I should be able to say $\{f_n\}$ converges on that given bound. That being said, I am having difficulty finding those bounds of differentiation on $\{f_n\}$, and I could use some help thinking through this part of the problem.
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$$\lim_{x \to \pm\infty} f_n(x) = 0$$
$$f_n'(x)=\frac{3-nx^2}{(3+nx^2)^2}$$
$$|f_n(x)| \le |f_n(\pm\sqrt{\frac{3}{n}})| = \frac{1}{6}\sqrt{\frac{3}{n}} \to 0 $$
$$||f_n||_{\infty} \to 0$$
So $f_n$ converge uniformly to $f$ with $f(x)=0$.
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