Is DxD -> R by e(z,w)=|(z-w)/(1-w'z)| a Metric Space?

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The discussion centers on proving that the function e: D x D -> R defined by e(z,w) = |(z-w)/(1-w'z)|, where D = {z in C | |z| < 1} and w' is the conjugate of w, constitutes a metric space. The primary challenge identified is demonstrating the triangle inequality, which is crucial for establishing the metric properties. Participants express confusion regarding the manipulation of the function and suggest the possibility of reclassifying the topic under Analysis rather than Differential Geometry or Topology.

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D={z in C | |z|<1}
e: DxD -> R by e(z,w)=|(z-w)/(1-w'z)| (here the w'=the conjugate of w, not sure how to insert a bar on top of the w). Show that this is a metric space. It's all pretty easy till the triangle inequality (as always though, right?) so that's all I need to focus on. I'm pretty lost on where to start for this one. I've tried several manipulations and even tried working with this all in a+bi form, and never really saw any good place to insert a +/-x and make an inequality. Any help?
 
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No help? Maybe this should be moved to Analysis instead of Diff Geom/Topology?
 

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