inequality with absolue value

Question

For every $x,y \in \mathbb{R}$

$$\dfrac{\mid x+y\mid}{1+\mid x + y \mid} \le \dfrac{\mid x \mid}{1+\mid x \mid} + \dfrac{\mid y \mid}{1+\mid y \mid}$$

Having hard time proving this statement . Please help !

Thanks

Answer 1

Note that $\frac{u}{1+u}$ is increasing in $u$, and $|x+y| \leq |x|+|y|$.

Hence

$\frac{|x+y|}{1+|x+y|} \leq \frac{|x|+|y|}{1+|x|+|y|}\leq \frac{|x|}{1+|x|}+\frac{|y|}{1+|y|}$