An oftcited theorem of Serre states that there are infinitely many geodesics between any two points in a closed Riemannian manifold. Could someone please provide an intuitive sketch of the proof?

The question having been reopened, I'll repost my comment as an answer: There is a paper by Nabutovsky and Rotman, available at http://www.math.toronto.edu/alex/morseoct12.pdf in which they give "a somewhat modernized sketch of the proof of Serre’s theorem given by A.Schwarz...." 

