Prove the inequality for x,y real numbers

Question

I'm trying to prove this inequality using the triangle inequality, but unfortunately haven't had much luck. I feel like I can rewrite the left hand side into something usable, but I don't know what to. The inequality is for x,y real numbers:

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

Any help or solution would be appreciated.

Answer 1

Consider $ \frac{|x+y|}{1+|x+y|} = 1-\frac{1}{1+|x+y|} \leq 1-\frac{1}{1+|x|+|y|+2|xy|}$ by the triangle inequality: adding an extra positive term on the bottom obviously will keep satisfying the inequality.

That equals:

$ \frac{|x|+|y|+2|xy|}{1+|x|+|y|+2|xy|} = \frac{|x|}{1+|x|} +\frac{|y|}{1+|y|}$

so we prove the required inequality.