Definition:    x is infinitely close to y, x » y, iff x -* y Î I

Theorem:       » is an equivalence relation on R*.

Proof:           (Reflexivity)    0 Î I Þ x -* x Î I.

                    (Symmetry)    x -* y Î I Þ y -* x Î I because of closure under ±*.

                    i.e. the negative of an infinitesimal is also an infinitesimal.

                    (Transitivity)   x -* z = (x -* y) +* (y -* z) Î I because of closure again.  

Theorem:       If u » v and x » y, then u +* x » v +* y, and -*u » -*v

Proof:            (u +* x) -* (v +* y) = (u -* v) +* (x -* y) Î I because of ±* closure.

Theorem:       If u » v and x » y, and u, v, x, y Î F, then u .* x » v .* y

Proof:            (u .* x) -* (v .* y) = (u .* x) -* (u .* y) +* (u .* y) -* (v .* y)

                                                = u .* (x -* y) +* (u -* v) .* y

                                                Î I because of closure of I under multiplication by elements of F.

Theorem:       If ~ (u » v) and x Î F or y Î F then $q Î R x <* q <* r

Proof:           Suppose wolog 0 £* x <* y

                    ~(x » y) Þ $b 0 <* b <* y -* x

                    x Î F Þ $m Î Z* x <* mb

                    Take the least such m, then x <* mb <* y

                    This follows since (m-1)b £* x Þ mb £* x +* b <* y