I'm reading an article (http://arxiv.org/abs/gr-qc/0403075) which proves that a certain spacetime is geodesically complete. It does this by proving that the first derivatives fo all coordinates have finite bounds. My question is why this is enough. Is it just a simple ODE result? We know that the geodesic equation locally has a solution given an initial "position" and "velocity", by the basic existence result in ODEs. Hence if we show that this "velocity" is bounded everywhere then then the geodesic can extend indefinitely, because the geodesic equation can be solved everywhere. Is this right?