I am trying to prove the following corollary from the book Differential Geometry of Curves and Surfaces by Kristopher Tapp, p. 332.
If $S$ is a regular surface that is diffeomorphic to a cylinder and has Gaussian curvature $K<0$, then $S$ has at most one simple closed geodesic (up to reparametrization).
Proof is done with the assumption of the existence of two distinct simple closed geodesics. Their traces are denoted by $C_1,C_2$. Case (1) says that if $C_1$ and $C_2$ intersected in one point, this would contradict the uniquness of geodesics.
I handled Case (2) and Case (3) which are the intersection of multiple points and no intersection at all. However, I do not understand the contradiction with the uniquness of geodesics in Case (1). How could I show that?
I have found an answer saying that if two geodesics intersect in exactly one point, then they must be tangent at their point of intersection, which, by the uniqueness of geodesics, implies that they describe the same curve. Hence, contradiction.