König's Lemma/Proof 3

Theorem

Let $G$ be an infinite graph which is connected and is locally finite.

Then every vertex lies on a path of infinite length.

Proof

By Locally Finite Connected Graph is Countable, $G$ has countably many vertices.

Thus the result holds by König's Lemma: Countable.

$\blacksquare$

Axiom:Axiom of Countable Choice for Finite Sets

This theorem depends on Axiom:Axiom of Countable Choice for Finite Sets, by way of Locally Finite Connected Graph is Countable.

Although not as strong as the Axiom of Choice, Axiom:Axiom of Countable Choice for Finite Sets is similarly independent of the Zermelo-Fraenkel axioms.

As such, mathematicians are generally convinced of its truth and believe that it should be generally accepted.