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Following problem had post mathstack three years ago,and until now no one solve it.so I ask here,Thanks for you help.

let $$f(x,y)=\dfrac{|x-y|}{\sqrt{|x|^2+1}+\sqrt{|y|^2+1}}$$ show that $$f(x,y)+f(y,z)\ge f(x,z)$$ Background:

Show that $(X,d)$ is a metric space where $X =\Bbb C $ and the distance function is defined as: $$d(x,y) = \frac {2|x-y|}{\sqrt {1+|x|^2} + \sqrt {1 + |y|^2}}, \text{ for } x,y \in \Bbb C.$$

I have done the proof of the first two propositions for being a metric, but I'm having a problem in proving the triangle inequality.

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    $\begingroup$ Does this come from somewhere? This has been asked multiple times on math.se but it doesn't seem anyone says where it comes from $\endgroup$
    – user35370
    Commented Dec 6, 2018 at 2:28
  • $\begingroup$ @fedja: I withdrew my comment. And voted to reopen. But you can post your solution to MathSE as well. $\endgroup$
    – Alexandre Eremenko
    Commented Dec 7, 2018 at 1:00
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    $\begingroup$ @AlexandreEremenko OK, I removed my responses to your comment too as no longer relevant and did exactly as you suggested. The argument is not technically complicated at all, indeed, but it still took me a few hours to come up with it. :-) $\endgroup$
    – fedja
    Commented Dec 7, 2018 at 3:13

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