An information-theoretic lower bound is developed for the caching system studied by Maddah-Ali and Niesen. By comparing the proposed lower bound with the decentralized coded caching scheme of Maddah-Ali and Niesen, the optimal memory--rate tradeoff is characterized to within a multiplicative gap of 4.7 for the worst case, improving the previous analytical gap of 12. Furthermore, for the case when users' requests follow the uniform distribution, the multiplicative gap is tightened to 4.7, improving the previous analytical gap of 72. As an independent result of interest, for the single-user average case in which the user requests multiple files, it is proved that caching the most requested files is optimal.