  Mathematicians tend to use the phrase "arbitrarily close" to mean something
along the lines of "every neighborhood of a point intersects a set". Taking the
latter statement as a technical definition for arbitrarily close leads to an
alternative development of classic concepts in real analysis such as supremum,
closure, convergence and limits of sequences, closure, connectedness,
compactness, and continuity. The goal of this text is to provide readers with
an introduction to real analysis by taking deliberate steps to parse these
difficult concepts using arbitrarily close as the kernel.
