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