To prove that $\frac{|x-y|}{\sqrt{1+|x|^2}+\sqrt{1+|y|^2}}$ is a metric on $\mathbb{C}$

Question

I am practically clueless on how to prove the triangle inequality here. The case is evident when $|z|≤|x|$ and $|z|≤|y|$. But how do I prove it in general?

Any hint/answer would be much appreciated.