Simple Graph with Finite Vertex Set is Finite

Theorem

Let $G$ be a simple graph.

Suppose that the vertex set of $G$ is finite.

Then $G$ is a finite graph.

That is to say, its edge set is also finite.

Proof

Since $G$ is simple, it can have at most one edge between each two vertices.

Therefore, the mapping:

$v: \map E G \to \powerset {\map V G}, \map v e = \set {a_e, b_e}$

which assigns to each edge its endvertices $a_e, b_e$, is an injection.

From Power Set of Finite Set is Finite and Domain of Injection Not Larger than Codomain, it follows that $\map E G$ is finite.

Hence $G$ is finite.

$\blacksquare$