I heard this really neat elementary proof of the "Gauss-Bonnet Theorem" :

Let $S$ be a surface embedded in $\mathbb{R}^3$. Now take a triangulation of that surface, and approximate the surface with a triangular mesh ("flatten" the triangles).

Call $\delta(v)$ the *angle defect* at a vertex $v$, that is $2\pi$ minus the sum of the angles around the vertex $v$. Now, the Euler characteristic is defined at $\chi(S) = V - E + F$. Also, if we assume that the surface is without boundary then every face has $3$ bounding edges and every edge bounds $2$ faces, which means $2E=3F$, so $F=2(E-F)$.

Let's compute the sum of all the angle defects at all the vertices of the mesh in two different ways. First, sum the angle defect at each vertex, that is $\sum_{v} \delta(v)$. Second, for each triangle the sum of the angles is exactly $\pi$ since they are euclidean triangles in $\mathbb{R}^3$, so we can sum of the angles of all the faces and get $F\pi$, and subtract that from what the total angle sum would be if the surface were flat to get $2 \pi V - F \pi$. Using the formula for the Euler characteristic, we get

$\sum_{v} \delta(v) = 2 \pi V - \pi F$

$\sum_{v} \delta(v) = 2\pi (V - E + F) = 2\pi\chi(S)$

Now, to make that into the "real" Gauss-Bonnet Theorem I heard the argument "take an infinitesimal triangulation, then the angle defect becomes the curvature". Is there any way to make this rigorous with a reasonable amount of not too advanced tools? If so, can I get a reference?

Thanks!