Page "König's lemma" Paragraph 5

Start with any vertex v < sub > 1 </ sub >.

Every one of the infinitely many vertices of G can be reached from v < sub > 1 </ sub > with a simple path, and each such path must start with one of the finitely many vertices adjacent to v < sub > 1 </ sub >.

There must be one of those adjacent vertices through which infinitely many vertices can be reached without going through v < sub > 1 </ sub >.

If there were not, then the entire graph would be the union of finitely many finite sets, and thus finite, contradicting the assumption that the graph is infinite.

We may thus pick one of these vertices and call it v < sub > 2 </ sub >.