König's Lemma

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

Then every vertex lies on a path of infinite length.

Note
If the graph $G$ is assumed to be countably infinite, then the result will hold in pure Zermelo-Fraenkel set theory without any choice.

Linguistic Note
Note that the diacritics on the ö in the König of König's Lemma and on the ő of do not match. Unfortunately, this is how it is.