I started reading a Morse theory by Milnor and am not understanding something. I am reading the proof of Theorem on page 25: Let M be a compact manifold and f be a differentiable function on M with only two critical points, both of which are non-degenerate, then M is homeomorphic to a sphere. We may assume that 0 is mimimum and 1 is maximum of f. In the proof he says that by Morse lemma for small epsilon f^-1[0, epsilon], f^-1[1-epsilon, 1] are closed n-cell. I guess he is using the fact that on f^-1(1) and f^-1(0) the morse index is 0. But Is the fact obvious? I know 0 and 1 are minimum and maximum respectively. So Hessian is positive at f^-1(0) and negative at f^-1(1). But why is morse index 0 at these points?