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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!

  • 1
    See also some other duplicates: [Proof of uniform convergence on $\mathbb{R}$](http://math.stackexchange.com/questions/572842/proof-of-uniform-convergence-on-mathbbr) ; [Uniformly convergent](http://math.stackexchange.com/questions/549761/uniformly-convergent) ; [Let $f_n(x) =\frac{x}{1+nx^2}$ and what function does this sequence converge to?](http://math.stackexchange.com/questions/1548122/let-f-nx-fracx1nx2-and-what-function-does-this-sequence-converge-to) ; [Uniform convergence of functions (...)](http://math.stackexchange.com/questions/414432/uniform-convergence-of-functions-and-intervals) – 2017-01-30
  • 1
    ... [Does $f_n(x) = \frac{x}{1 + nx^2}$ converge uniformly for $x \in \mathbb{R}$?](http://math.stackexchange.com/questions/1157218/does-f-nx-fracx1-nx2-converge-uniformly-for-x-in-mathbbr) ; [Does the sequence$ f_n(x)=\dfrac{x}{1+nx^2}$ converge uniformly on $\mathbb{R}$?](http://math.stackexchange.com/questions/1045996/does-the-sequence-f-nx-dfracx1nx2-converge-uniformly-on-mathbbr) – 2017-01-30

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