A well-known theorem by J.P. Serre asserts that for every pair of points on a closed Riemannian manifold there exist infinitely many distinct geodesics connecting these points. We prove that given a pair of points for every m there exist at least m geodesics of length <4nm^2d connecting these points, where n is the dimension of the manifold, and d is its diameter. Our proof provides some new information about Morse landscapes of the length functional on loop spaces. (Joint work with Regina Rotman.)