Inner angles in polygons

Question

A point of a polygon is called convex if it's inner angle is less than 180 degrees. Prove that in every simple polygon there is at least one convex point.

Answer 1

The sum of the inner angles of a simple polygon with $n$ sides is $(n-2)\pi$.

If all inner angles were greater than or equal to $\pi$, then that sum would be at least $n\pi$, which it isn't. Hence, at least one inner angle is less than $\pi$.

This argument actually implies that there are at least two convex vertices.