Suppose is any curve in the upper half-plane (the hyperbolic plane where ) connecting and . Prove that the hyperbolic length of is at least .
I know that if , then the geodesic that connects the two is simply the Euclidean line segment between the two points, and if , then the geodesic connecting the two points are arcs of Euclidean semicircles centered on the x-axis. Other than that, I really don't know where to begin. Thanks.