A Proof of Euler's Formula

Euler's formula states that for a map on the sphere, , where is the number of vertices, is the number of faces, and is the number of edges. This Demonstration shows a map in the plane (so the exterior face counts as a face). The formula is proved by deleting edges lying in a cycle (which causes and to each decrease by one) until there are no cycles left. Then one has a tree, and one can delete vertices of degree one and the edges connected to them until only a point is left. Each such move decreases and by one. So all the moves leave unchanged, but at the end and are each 1 and is 0, so must have been 2 at the start.