## Upper half-plane

Suppose $\gamma$ is any curve in the upper half-plane (the hyperbolic plane where $y>0$) connecting $(x,y)$ and $(x',y')$. Prove that the hyperbolic length of $\gamma$ is at least $|\ln (y',y)|$.

I know that if $x = x'$, then the geodesic that connects the two is simply the Euclidean line segment between the two points, and if $x\ne x'$, 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.