Geometric interpretation of Gaussian curvature.

Question

We have the following result from Do Carmo book of differential Geometry: "Let $p$ be a point of a surface $\Sigma$ such that the Gaussian curvature $K(p) \neq 0$, and let $V$ be a connected neighborhood of $p$ where $K$ doesn't change sign. Then $$ K(p)= \lim_{A \rightarrow 0} \frac{A'}{A} ,$$ where $A$ is the area of a region $B \subset V$ containing $p$, $A'$ is the area on the image of $B$ by the Gauss map $N: \Sigma \rightarrow S^2$, and the limit is taken through a sequence of region $B_n$ that converges to $p$, in the sense that any sphere around $p$ contains all $B_n$, for $n$ large enought". How can I prove, using this result, that is $\Sigma$ is a compact surface $$ \int_{\Sigma}|K| \ge 4 \pi \,\,\,? $$