I've seen in the answers to a few different questions here on the Mathematics Stack Exchange that one can clearly do mathematical induction over the set $\mathbb{R}$ of all real numbers. I am, however, having quite a difficult time understanding how the methods described in both those questions' answers and some reference materials to which they link. In particular, I can't seem to figure out exactly how the techniques described therein parallel the methods codified in the axiom of induction for use when doing mathematical induction over the set $\mathbb{N}$ of all natural numbers.

If somebody would be so kind as to provide me with a more detailed explanation of how to do mathematical induction over the set $\mathbb{R}$ of all real numbers within about the next day or so, then I would be very grateful! The answer should be understandable by any beginning calculus student who also has a rudimentary understanding of set theory and mathematical logic. I've provided links to both the relevant questions and whatever reference material mentioned in them that seemed like good leads when I found them no matter how inscrutable they might have been at the time.

Questions About Induction Over the Real Numbers:

- Induction on Real Numbers
- Is it possible to use mathematical induction to prove a statement concerning all real numbers, not necessarily just the integers? [duplicate]
- Extending a theorem true over the integers to reals and complex numbers

Question-Derived Reference Material:

- 'The Instructor's Guide to Real Induction' by Pete L. Clark

P. S.: I also have the following follow-up questions: