Showing that $x \sin\left(\frac1x\right) - y \sin\left(\frac1y\right)\leq |x+y|$

Question

This is a part of the solution in my textbook but I don't get it. Can somebody explain how this is true.

$$x \sin\left(\frac1x\right) - y \sin\left(\frac1y\right)\leq |x+y|$$

Answer 1

$-1 \leq \sin \left (\frac 1 x \right ), \sin \left ( \frac 1 y \right ) \leq 1,$ for all $x,y \neq 0.$ So for $x,y > 0$ $$-x - y \leq x \sin \left ( \frac 1 x \right ) - y \sin \left ( \frac 1 y \right ) \leq x+y.$$

For other values of $x$ and $y$ the inequality can be similarly obtained.