Non-standard analysis is that branch of mathematics that is concerned with analysis using the non-Archimedean ordered field of hyperreal numbers. It can be seen as the use of model theory to study analysis. Since studying the saturated model of a theory is easier than studying other models, non-standard analysis studies the saturated model of theories with many symbols thrown in to make sure results are applicable.
One kind of elements in the saturated model are infinitesimals. It is consistent for a positive non-standard real number to be smaller than any element of { 1/n | n in N }; thus, there is a positive non-standard real number smaller than all of these. In fact, there is a whole ideal of non-standard real numbers. If we start from the rationals, rather than the real numbers, and divide the ring of non-standard 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, epsilon-delta, analysis. For example, proving that the composition of continuous functions is continuous is much easier in a non-standard setting.
There are not many results proven first with non-standard 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.
Non-standard analysis was introduced by the mathematician Abraham Robinson in 1966 with the publication of his book Non-standard Analysis.
H. Jerome Keisler has written a practical elementary calculus text that applies Robinson's method. Elementary Calculus: An Approach Using Infinitesimals by H. Jerome Keisler http://www.math.wisc.edu/~keisler/calc.html
