Question

Infinitesimals are fascinating!

Answer 1

To give you some feedback, it would be good for you to have a clear picture in mind of *who* you are writing for, and what you want them to leave with a new understanding of. Somebody with no background in the hyperreals would struggle to get much information out of this, and somebody with some light level of understanding wouldn't learn anything new.

The example that you gave isn't particularly clear to me at all. In a kind of "introductory" text, you should try to at least start with a very simple, grounded, example of how they work or what makes them useful, formalising the notion of a derivative being f(x+dx) =f(x) + f'(x)dx, for example, would be a good introduction that people can grasp, and provides historical context for their definitionm

Also, a minor point, you say epsilon<x for all real x, but judging by the argument I presume you mean epsilon <x for all real **positive** x.

Answer 2

You mention the construction of the hypperreals through adjoining an element &epsilon; and applying the transfer principle. I’d argue this is not the most intuitive construction. It might be worth giving a simplified version of the construction through ultraproducts as a sort of continuation of the idea of Cauchy completion. Instead of going into ultrafilters you can just use the measure interpretation and say if something happens infinitely often in a sequence then it happens for sure in the hyperreals.