Möbius tranformation

Question

We define a Möbius transformation through: $$z\rightarrow \frac{az+b}{cz+d}, ad-bc\neq0, a,b,c,d\in \mathbb{C}$$ and extend to the Riemann sphere as follows: if $c=0$, $T(\infty)=\infty$, and if $c\neq0$, $T(\infty)=a/c$ and $T(-d/c)=\infty$. Show that the Möbius transformations are continuous in that area with the chordal metric. We define the chordal metric thus: $$d_{c}(z_{1},z_{2})= \begin{cases} \frac{2|z_{1}-z_{2}|}{\sqrt{1+|z_{1}|^2}\sqrt{1+|z_{2}|^2}}& \text{if $z_{1},\,z_{2}\neq \infty$}\\ \frac{2}{\sqrt{1+|z_{1}|^2}} & \text{if $z_{2}= \infty$.} \end{cases}$$