In this paper a proof of the
Jordan curve theorem will be presented. Some familiarity with the basic notions of
nonstandard analysis is assumed. The rest of the paper is selfcontained except for
some standard theorems about polygons.
The theorem will be proved in what ought to be a natural way: by approximation
by polygons. This method is not usually found in the standard proofs since the
approximating sequence of polygons is often unwieldly. But by using nonstandard
analysis, one can approximate a Jordan curve by a single polygon that is
infinitesimally close to the curve. This allows types of reasoning which are extremely
difficult and unnatural on sequences of polygons.
