Is $\frac{\vert x+y\vert}{1+\vert x+y\vert}\leq\frac{\vert x\vert}{1+\vert x\vert}+\frac{\vert y\vert}{1+\vert y\vert}$ true for all $x,y\in\mathbb{R}$?

If not, how can I prove that $\int\frac{\vert f-h\vert}{1+\vert f-h\vert}\leq\int\frac{\vert f-g\vert}{1+\vert f-g\vert}+\int\frac{\vert g-h\vert}{1+\vert g-h\vert}$?
I tried C-S:
$$\sum\_{cyc}\frac{|x|}{1+|x|}=\sum\_{cyc}\frac{x^2}{|x|+x^2}\geq\frac{(x+y)^2}{x^2+y^2+|x|+|y|}.$$
Thus, it's enough to prove that
$$\frac{(x+y)^2}{x^2+y^2+|x|+|y|}\geq\frac{(x+y)^2}{|x+y|+(x+y)^2}$$ or
$$|x+y|+(x+y)^2\geq x^2+y^2+|x|+|y|,$$ which is wrong for $(x,y)=(1,-1).$