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This article is about mathematics. For housing estates, see overspill estate. In non-standard analysis, a branch of mathematics, overspill (referred to as overflow by Goldblatt (1998, p. 129)) is a widely used proof technique. It is based on the fact that the set of standard natural numbers N is not an internal subset of the internal set *N of hypernatural numbers. By applying the induction principle for the standard integers N and the transfer principle we get the principle of internal induction: For any internal subset A of *N, if 1 is an element of A, and for every element n of A, n + 1 also belongs to A, then A = *N If N were an internal set, then instantiating the internal induction principle with N, it would follow N = *N which is known not to be the case. The overspill principle has a number of useful consequences: The set of standard hyperreals is not internal. The set of bounded hyperreals is not internal. The set of infinitesimal hyperreals is not internal. In particular: If an internal set contains all infinitesimal non-negative hyperreals, it contains a positive non-infinitesimal (or appreciable) hyperreal. If an internal set contains N it contains an unbounded element of *N. Example These facts can be used to prove the equivalence of the following two conditions for an internal hyperreal-valued function ƒ defined on *R. and The proof that the second fact implies the first uses overspill, since given a non-infinitesimal positive ε, By overspill a positive appreciable δ with the requisite properties exists. These equivalent conditions express the property known in non-standard analysis as S-continuity of ƒ at x. S-continuity is referred to as an external property, since its extension (e.g. the set of pairs (ƒ, x) such that ƒ is S-continuous at x) is not an internal set. References R. Goldblatt (1998). Lectures on the hyperreals. An introduction to nonstandard analysis. Springer.