König's Lemma/Proof 3
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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.