In my studies of differential geometry from do Carmo's book, I have come across a very nice claim which states that a regular compact surface has an elliptic point that is a point with positive Gaussian curvature I have read the proof and it said that looking at the normal sections at a point where the surface and a sphere are tangent, we see that the normal curvatures at this point of the surface is greater than those for the sphere known to be positive. I have no idea how they reached this conclusion as I do not see any obvious relation between normal sections of the sphere and surface aside from the surface and sphere being tangent, and I certainly would appreciate the help explaining it. I have attached the proof here highlighting the conclusion I do not know how to deduce, I thank all helpers
The surface is contained inside the sphere, but the tangent plane is outside the sphere, except at $p$. Therefore as you leave $p$ in any direction, the surface will curve away from the tangent plane at least as fast as the sphere does.