Prove that the sequence {$k_n$}$_{n=1}^\infty$, defined by $$k_n (x) = \frac{x}{1+nx^2}$$ for all x $\in \mathbb{R}$ and each positive integer n, converges uniformly on $\mathbb{R}$.
Any help would be appreciated.
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Here is a technique. – Mhenni Benghorbal Nov 19 '13 at 05:34
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@MhenniBenghorbal this method seems like it would require splitting it up into three intervals- from -infinity to the infimum, infimum to supremum, and supremum to infinity. is that the case, or is there a way to do it in one? – Nick Nov 19 '13 at 05:42
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Note that, you need to find the following
$$ \sup_{-\infty<x<\infty} \Bigg| \frac{x}{1+n x^2} - 0\Bigg|. $$
Let
$$ g(x)= \frac{x}{1+n x^2}.$$
Using the derivative technique, the max of $g$ is achieved at $x=\frac{1}{\sqrt{n}}$ (by the second derivative test) and it's given by
$$ g(1/\sqrt{n})=\frac{1}{2\sqrt{n}} \implies \frac{1}{2\sqrt{n}}< \epsilon $$
Mhenni Benghorbal
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