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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!


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

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