My given question is:
In hyperbolic space, given a geodesic $L$ and a point $p$ not lying on $L$, show there is an infinite number of geodesics through $p$ which do not intersect $L$.
The "model" for hyperbolic space is the upper half plane $H=\{(x,y):y>0\}$ where in this model, distance decreases as we move up. In this way, I can understand how it's true in my head, but I'm not sure how to start to prove this.
Where is a good place to start proving this?
Sketch of a possible approach:
Show (or know) that a geodesic in the hyperbolic space is the equivalent in the the upper half plane model of a vertical ray or of a semicircle centred on the $x$-axis
Consider the three possibilities that $L$ is a vertical ray with $p$ not on $L$, or $L$ is a semicircle with $p$ "inside" the semicircle, or $L$ is a semicircle with $p$ "outside" the semicircle
For each possibility, show there is an interval on the $x$-axis where all semi-circles centred on that interval and passing through $p$ do not intersect $L$
Draw your conclusion