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

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