Let $z$ and $w$ be complex values such that $|z| \leq 1, |w| \leq 1$ and $|\bar{z}w| \neq 1$, prove that $$\left|\frac{w-z}{1-\bar{z}w}\right| \leq 1$$
I was trying to solve using Schwarz inequality but I got nowhere, first i noticed that $|wz| \leq 1$ and use that $|w-z| \leq 2$ then
$$ \left|\frac{w-z}{1-\bar{z}w}\right| = \frac{|w-z|}{\left|1-\bar{z}w\right|} \leq \frac{2}{\left|1-\bar{z}w\right|} $$
but now i need to decrease $\left|1-\bar{z}w\right|$ to improve the quotient but i only remember that $\left|1-\bar{z}w\right| \geq 1 -1 = 0$.