We give a nonstandard treatment of the notion of ends of proper geodesic
metric spaces. We then apply this nonstandard treatment to Cayley graphs of
finitely generated groups and give nonstandard proofs of many of the
fundamental results concerning ends of groups. We end with an analogous
nonstandard treatment of the ends of relatively Cayley graphs, that is Cayley
graphs of cosets of finitely generated groups.Comment: 23 page