Show that $\left|\frac{\alpha - \beta}{1-\bar{\alpha}\beta}\right| < 1$ when $|\alpha|,|\beta| < 1$

Hint: $$ \left| \frac{\alpha-\beta}{1-\bar\alpha\beta}\right| < 1 \Leftrightarrow |\alpha-\beta|^2 < |1-\bar\alpha\beta|^2. $$

Expand both sides, remembering that $|z|^2 = z\bar z$ and simplify. That should get you where you want to be after some algebra.