Prove that ${f_n}$ converges uniformly to $f:\mathbb{R}→\mathbb{R}$

Question

My question is the following:

$${f_n(x)}= \frac1n \arctan({n^2x})$$ Prove that ${f_n}$ converges uniformly to a continuous function $f:\mathbb{R}\to\mathbb{R}$.

Also if proven, how would I create a formula for $f(x)$. I know that by definition it converges if there exists $\epsilon >0$ such that $|f(x) - f_n(x)|< \epsilon$.

Answer 1

$|\arctan(x)|$ is bounded by $\pi/2$ for all $x \in \mathbb R$, so when $n$ tends to $\infty$, $f_n$ goes to zero.

You should prove, given an $\varepsilon >0$, $\exists N \in \mathbb N$ such that $|f_n(x) - 0| < \varepsilon$ for all $n \geq N$ and $x \in \mathbb R$.