How to prove this inequality $x,y\in\Bbb R$, $|x|<1,|y|<1$ show that $\bigg|\frac{x-y}{1-xy}\bigg| < 1$ (and similar ones)

Question

I have to show that the inequality below is true, i tried some thing but got stuck, i tried to eliminate the absolute value $-1<\frac{x-y}{1-xy}<1$ and then solve for $x$ and $y$ with no luck...i do not want the answer to this problem but at least a method for solving this kind of exercises.

$$x,y\in\Bbb R\ , |x|<1,|y|<1 \ \ \text{show that} \ \ \bigg|\frac{x-y}{1-xy}\bigg| < 1.$$

Answer 1

Observe:

$|x - y|^2 < |1 - xy|^2 \Leftrightarrow (1 - |x|^2)(1 - |y|^2) > 0$

Answer 2

$$\left|\frac{x-y}{1-xy}\right|<1\Longleftrightarrow |x-y|<|1-xy|\Longleftrightarrow x^2-2xy+y^2<1-2xy+x^2y^2\Longleftrightarrow$$

$$\Longleftrightarrow (x^2-1)(y^2-1)>0$$

and since the last inequality above is trivially true we're done