=Richard Schroeppel Could we be a bit more specific about what a Path or Cycle on an infinite graph is supposed to mean?

Exactly the question! Now I see why I've been very confused by this part of the discussion. If we mean by Cycle simply something like "there exists a connected subgraph with all nodes valence 2", and by Path "ditto, but allowing at most two nodes of valence 1", then I see no contradiction with the examples (and the desired proof becomes trivial). Only if we insist that everything be FINITELY connected are these infinite hairpins a problem. But then it kind of begs the question whether the infinite graph itself is "connected" in the first place.