Divergence of curl is zero (coordinate free approach)

The problem is, as @Mark S. commented a while ago, very badly written. I am going to reiterate the suggestions of that comment as an answer, because this is an important technique.

Yes, surely the professor intends to assume $\vec F$ is $C^2$. The gist of the argument is a common one in math and physics. If $\nabla\cdot(\nabla\times\vec F)$ is not everywhere zero, then at some point $P$ it is nonzero, hence — say — positive. By continuity, it is positive on a small ball around $P$. Now you get a contradiction by taking your little surfaces $S_1$ and $S_2$ within that ball.