Construct a homotopy of geodesics from "common perpendicular" $h$ to a geodesic $c'$ from $c(0)$ to $c(1)$ with the ends are on A and B. (This is possible since curvature is negative.) In this process, the number of self intersections stays constant; i.e. $c'$ is simple. Thus, we get two geodesics $c$ and $c'$ with common ends. Since curvature is negative, $c\sim c'$ implies $c=c'$, which is not possible if $c$ is non-simple.
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