Page "König's lemma" Paragraph 7

Continuing in this fashion, an infinite simple path can be constructed by mathematical induction.

At each step, the induction hypothesis states that there are infinitely many nodes reachable by a simple path from a particular node that does not go through one of a finite set of vertices.

The induction argument is that one of the vertices adjacent to satisfies the induction hypothesis, even when is added to the finite set.