One kind of elements in the saturated model are infinitesimals. It is consistent for a positive real number to be smaller then any element of { 1/n  n in N };thus, there is a positive nonstandard real number smaller than all of these. In fact, there is a whole ideal of nonstandard real numbers. If we start from the rationals, rather then the real numbers, and divide the ring of nonstandard finite rational numbers by the ideal of the infinitesimal rational numbers, we get a field (because it is a maximal ideal)  the field of real numbers. This sometimes gives easier ways to prove results which are hard work in classical, epsilondelta, analysis. For example, proving that the composition of continuous functions is continuous is much easier in a nonstandard setting.
There are not many results proven first with nonstandard analysis. One of them is the fact that every polynomially compact linear operator[?] on a Hilbert space has an invariant subspace, proven 5 years before classic functional analysis techniques were developed that deal with such problems.
Nonstandard analysis was introduced by the mathematician Abraham Robinson in 1966 with the publication of his book Nonstandard Analysis.
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