  We present a viscosity approach to the min-max construction of closed
geodesics on compact Riemannian manifolds of arbitrary dimension. We also
construct counter-examples in dimension $1$ and $2$ to the
$\varepsilon$-regularity in the convergence procedure. Furthermore, we prove
the lower semi-continuity of the index of our sequence of critical points
converging towards a closed non-trivial geodesic.
