Prove that Gauss map on M is surjective Let $M$ be a closed, orientable, and bounded surface in $\mathbb{R}^3$.
(a) Prove that the Gauss map on $M$ is surjective.
(b) Let $K_+(p) = \max \{0, K(p)\}$.  Show that
$$
\int K_+dA \ge 4\pi.
$$
in the area of $M$.  Do not use the Gauss-Bonnet Theorem to prove this.

For a) there is a simple argument (due to Hilbert IIRC). Pick any unit vector $v$ and imagine a plane with normal vector $v$ with distance $t$ from the origin (imagine $t$ large enough so it is further away from the origin than $M$).
Reduce $t$ until the plane first touches the surface. The point on $M$ where it first touches is a point on $M$ whose image under the Gauss map is $v.$ To make this idea rigourous, you will want to formalize the notion of distance between $M$ and the plane. The compactness of $M$ will ensure the infimum is a minimum.
