What is the proof of infinite number of parallel lines in hyperbolic geometry?

Question

I know that parallel axiom in Hyperbolic geometry is that there at least two parallel lines to line $a$ through a given point $A$. But as I know it can be proven that there are infinitely many of them, what is the proof ?

Answer 1

Suppose you have the two lines through $A$, that converge on the two ends of $a$.

A point $B$ that lies between these convergent lines, will create a line $AB$, that crosses the two convergent lines at $A$, and so will always be opposite one of the convergent lines to $a$. Thus it never crosses $a$.