Proof of uniform convergence on $\mathbb{R}$

Question

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.

Answer 1

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

$$ g(1/\sqrt{n})=\frac{1}{2\sqrt{n}} \implies \frac{1}{2\sqrt{n}}< \epsilon $$