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Arbitrarily Close

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.