More non-standard analysis (infinitesimal calculus). The problem is from the free online book "Elementary Calculus: An Infinitesimal Approach" given to me earlier today by Plato. Please let me know if this proof is solid.

Given: (*)

are hyperreal numbers such that

.

Prove that if a and b are finite, then . (≈ means "are infinitely close" in this context).

1. Assume that a and b are finite, and let , , and be infinitesimal.

2. Since and , then and .

3. Since a and b are finite and and , then and are finite.

4. Since and are finite, then

5. From 2, =

=

.

6. The sum is infinitesimal according to the rules for infinitesimal and finite numbers. Let . Then

7. Since is infinitesimal, it follows that ■