Given $x= (x_1, x_2)$, and the following state equation: $$\dot{x_1} = -x_1+\frac{2x_2}{1+{x_2}^2}$$ $$\dot{x_2} = -x_2+\frac{2x_1}{1+{x_1}^2}$$ Show that the solution satisfies the inequality $$||x(t)||_2 \leq e^{-t}||x(0)||_2+\sqrt{2}(1-e^{-t})$$
I could only notice one thing which is that $(1+{x_i}^2)$ has derivative $2x_i$. Not sure if it is helpful. The question has a hint that asks to use the comparison lemma, but I'm not sure how to solve this. Any help is appreciated.
$\textbf{Hint}:$ You have the inequality
$$-1 \leq \frac{2z}{1+z^2}\leq 1$$
for all $z\in\Bbb{R}$