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For all $n\geq 1$, let $f_n: \mathbb{R}\rightarrow \mathbb{R}$ be defined by $f_n(x) = \frac{x}{1+nx^2}$. Show that the sequence of functions ${f_n}$ converges uniformly to some function $f:\mathbb{R}\rightarrow\mathbb{R}$.

My attempt: I think as $n\rightarrow\infty$, $f_n(x) = \frac{\frac{1}{n}x}{\frac{1}{n}+x^2}$ goes to $\frac{1}{nx}$, which goes to $0$ (is this true? I'm having trouble rigorously justifying it).

Then I want to show that for all x, $\sup_{x\in\mathbb{R}}|f_n(x)|\rightarrow 0$ as $n\rightarrow\infty$. This I'm having trouble with, as the reasoning seems similar to above, but I'm not positive.

Any help appreciated!

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marked as duplicate by Winther, TonyK, Shailesh, Cyclohexanol., user91500 Jan 30 '17 at 5:54

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