Let we have $C$ the set of complex numbers and $z_1 , z_2 \in C $ we have $Z_1 , Z_2 \in S$ correspond on riemann sphere and we will define : $$ d(Z_1,Z_2)=\frac{2|z_1-z_2|}{\sqrt{1+|z_1|^2} \sqrt{1+|z_2|^2} } $$ How can I prove $d$ is a distance
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Are $z_1, z_2$ images of $Z_1, Z_2$ under the stereographi projection? – gary Apr 12 '15 at 03:51
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@gary yes man – Functional analysis Apr 12 '15 at 04:02
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Then , in general, if d(x,y) is a metric in X and h: X-->Y is a homeomorphism, then d(h(x),h(y)) is a metric in Y, i.e., metrics pullback by homeomorphisms. – gary Apr 12 '15 at 04:08
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Hint: Calculate the Euclidean distance between $Z_1$ and $Z_2$ then express it in terms of $z_1$ and $z_2$.
Tim Raczkowski
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