Geodesic crosses every parallel in a surface of revolution.
Solution 1:
If the geodesic fails to cross every parallel, then either (a) it intersects a "maximum/minimum" parallel $u=u_0$ tangentially or (b) it approaches a parallel $u=u_0$ asymptotically. In either event, we must have $\dot u$ either equal to or approaching $0$ on the geodesic. This contradicts the positive lower bound on $|\dot u|$ given in the text.