Consider a closed, simple, smooth curve $C$. Can someone help me prove the following:
For all point in $C$ and their respective neighborhoods, the radius of curvature cannot be infinite.
I basically came up at this point while trying to find some properties of closed curves. But, as one might've understood already that i want to prove that a closed simple plane curve MUST have either a vertex, or a curved region(radius of curvature is finite). Can someone give some hints as to how to prove this, or just disprove this claim?