- #1

sal

- 78

- 2

I have no problem with the conventional proof. However, in Henle&Kleinberg's Infinitesimal Calculus, p. 123 (Dover edition), they give a nonstandard proof, and they use the

*reals to do it. I don't understand why that's needed.*

__hyper__hyperThe proof is very short; here's an outline:

Let '~' mean

*infinitesimally close to*and '~~' mean

*hyperinfinitesimally close to*(i.e., infinitesimal as viewed from within the hyperreals, using the hyperhyperreal system constructed on top of the hyperreals).

Given a functions

*f*

_{n}→ (uniformly)

*f*, and given any real

*x*,

**THEN**for any

*h*~~

*x*, and for any infinite

*N*, we'll have:

f

_{N}(x) ~~ f

_{N}(h) (by assumption),

f

_{N}(x) ~ f(x) (by assumption),

and f

_{N}(h) ~ f(h) (by assumption).

Then we have f(x) ~ f

_{N}(x) ~~ f

_{N}(h) ~ f(h), and so f(x) ~ f(h), and

*f*is continuous at

*x*.

I don't understand why the proof doesn't work equally well if we just use the hyperreals, and replace the '~~' with '~' everywhere. Any clue would be appreciated.