Please, I need help with this exercise of Complex Analysis, can anybody help me? Thank you so much.
Let be $\textbf{D}$ the open unit disc in the complex plane. Let be $a$, $b$ $\in$ $\textbf{D}$.
Prove that $d(a,b):=\left|\frac{a-b}{1-\overline{a}b}\right|$ is a distance in $\textbf{D}$ such that $d(a,b)<1$ for all $a$, $b$ $\in$ $\textbf{D}$.
Is $(\textbf{D},d)$ a complete metric space?
I tried this:
$d(a,b):=\left|\frac{a-b}{1-\overline{a}b}\right|=\frac{\left|a-b\right|}{\left|1-\overline{a}b\right|}\leq \frac{\left|a\right|+\left|b\right|}{\left|1-\overline{a}b\right|}=\frac{\left|a\right|+\left|b\right|}{\left|\overline{1-a\overline{b}}\right|}=\frac{\left|a\right|+\left|b\right|}{\left|1+a\overline{b}\right|}=\frac{2}{\left|1+a\overline{b}\right|}$
But I don't know how to continue. And I suppose that I have to use a Cauchy sequence to prove that $(\textbf{D},d)$ is complete.