Let n be the largest number of edges on any face of the polyhedron,
and let F be a face with n edges.
Each of these edges is adjacent another, distinct face of the polyhedron.
Therefore, the polyhedron has at least n+1 faces, each of which has between
1 and n edges (in fact, between 3 and n edges).
By the pigeonhole principle, two of those faces must have the same
number of edges.