Gaussian Curvature at the point where all curves passing through the point have curvature greater than C

Question

Let S be a regular surface and let p $\in$ S. Let $C > 0$ and suppose that every regular curve $ \alpha :(−\epsilon,\epsilon) \rightarrow S$ with $\alpha(0)=p$ has curvature at least C at p. Show that the Gaussian Curvature at p, $K(p) \ge C^{2}$.