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.
Prove that if a and b are finite, then . (≈ means "are infinitely close" in this context).
are hyperreal numbers such that
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
7. Since is infinitesimal, it follows that ■