Prove that the as Zn approached infinity, the Chordal Metric of Zn and zero approached zero.

Question

We've been asked this problem to prove that as $z_n\to \infty$, the chordal metric $\rho(z_n,\infty)\to 0$. Where $\rho(z_1,z_2)= d(z_1',z_2')$ where $z_1'$ and $z_2'$ are points on the Riemann Sphere, and $\rho$ is a metric on the complex plane. I am completely lost on how to approach this problem, besides trying to approach from making a point on the sphere and trying to come up with a parametric representation to it, but still has no clue how to prove this.

Answer 1

Carrying on the computations in $$\rho(z_1,z_2)=d(z_1',z_2')$$

(considering that $z_1'=f(z_1);z_2'=f(z_2)$, where $f$ is the inverse of the stereographic projection)

we get

\begin{align}\rho(z_1,z_2)&=\frac{|z_1-z_2|}{\sqrt{1+|z_1|^2}\sqrt{1+|z_2|^2}}\\ \rho(\infty,z_2)&=\frac{1}{\sqrt{1+|z_2|^2}}\end{align}

And thus the result follows.

Alternatively, one may note that the chordal metric is invariant under inversion, and thus reduce the problem to $\rho\left(\frac{1}{z_n},0\right)\to 0$, which is easily solvable