An oft-cited 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?



Proof of a theorem of Jean-Pierre Serre on geodesics of closed Riemannian manifolds
The question having been reopened, I'll re-post 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...."


