Uniform convergence for $x\arctan(nx)$

Question

I am to check the uniform convergence of this sequence of functions : $f_{n}(x) = x\arctan(nx)$ where $x \in \mathbb{R} $. I came to a conclusion that $f_{n}(x) \rightarrow \frac{\left|x\right|\pi}{2} $. So if $x\in [a,b]$ then $\sup_{x \in [a,b]}\left|f_n(x)- \frac{\left|x\right|\pi}{2}\right|\rightarrow 0$ as $n\to\infty.$ Now, how do I check the uniform convergence?

$$\sup_{x\in\mathbb{R}}\left|f_n(x)-\frac{\left|x\right|\pi}{2}\right| = ?$$

Thanks in advance!

Answer 1

Hint: Use the fact that $$\arctan t+\arctan\left(\frac 1t\right)=\frac{\pi}2$$ for all $t>0$ and that $f_n$ are even.