6. (8 points) Suppose that we already have the following theorem.

If $G$ is a connected planar simple graph with $e$ edges and $v$ vertices, where $v \ge 3$, then $e \le 3v - 6$.

Using this theorem to prove that if $G$ is a connected planar simple graph with at least three vertices, then $G$ has a vertex of degree not exceeding five.