In this article, the author gives a new proof of T. Aubin's result to the effect that the total scalar curvature functional achieves its infimum in a given conformal class of metrics provided this infimum is less than its value on the standard sphere. The method is more functional analytical in spirit than Aubin's. It relies on modifying a minimizing sequence to a converging one by steepest descent arguments. Non-triviality of the minimum is established by using the sharp Sobolev inequalities also due to T. Aubin. An extension to non-compact complete manifolds is considered at the end of the article. Stronger assumptions are then needed to ensure existence of a minimum.