Let m be a finitely additive measure on the set N of positive integers such that:
(The existence of m requires the Axiom of Choice.)
Now, let S be the set of all sequences {a_n} of real numbers.
Define an equivalence relation E by {a_n} E {b_n} iff m{n: a_n = b_n} = 1.
Now, define *R = S/E.
Writing <a_n> as the equivalence class of {a_n}, define addition, muliplication and ordering in *R by
<a_n> + <b_n> = <a_n + b_n>
<a_n><b_n> = <(a_n)(b_n)>
<a_n> < <b_n> iff m{n: a_n < b_n} = 1
(This can easily be shown to be well-defined.)
Identify a real number b with the equivalence class <b,b,b,b,.....>
Define x in *R to be infinitesimal iff -a < x < a for all positive real numbers a.
Then <1/n> = <1, 1/2, 1/3, 1/4, 1/5, ....> is infinitesimal. So are 0 and <1/n^2>.
0 < <1/n^2> < <1/n>
This page is adapted from Lindstrom's paper, An invitation to NSA (in NSA and its applications, CUP 1988).