We construct real numbers via Cauchy sequences, utilizing the Axiom of Countable Choice, which can be justified by appealing to the Proof Interpretation of logic and the construction of the natural numbers. Equality of real numbers is generally undecidable, so we look for positive concepts. The ordering a < b is defined by a positive lower bound on distance; the classical law of trichotomy then fails. The positive relation of apartness comes to replace inequality. The classical topological concepts like supremum and infimum are shown to be applicable only for sets satisfying the constructive property of being “located”.