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Here is my rather simple idea. I will treat the set of real numbers as a set of discrete continuities, each separated by an Epsilon ball that tends to 0.

So, let's say P(b) is true. We then assume P(k) is true, and prove that P(k+e) is true, where e goes to zero.

I just want to know if this is a valid technique or not because our teacher said that mathematical induction can only be applied to discrete structures, but I see no difficulty in treating a continuous system as a set of infinitesimal discrete quantities. Mak

Saikat
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    Google 'real induction'. There are notes by Pete Clark. – Pedro Jan 21 '16 at 00:43
  • then you will never reach to P(b+1) for example. Actually, you just have fun around epsilon neighborhood around b, which(epsilon) by the way tends to 0. – Mihail Jan 21 '16 at 00:45
  • Wow I have never heard about that! Thanks @PedroTamaroff – Maffred Jan 21 '16 at 00:49
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    By the way, you should work a bit on your exposition. For example, you wrote: "So, let's say P(b) is true. We then assume P(k) is true, and prove that P(k+e) is true, where e goes to zero"... And then what? What is it that you're trying to prove? That $P(x)$ holds for every real number $x$, for example? Also, "prove that P(k+e) is true, where e goes to zero" is too vague to be useful: what do you mean by that, precisely? – A.P. Jan 21 '16 at 01:01
  • The approach to real induction explained in the paper attached to this answer by Bill Dubuque is most similar to what you're hinting at here. I'm voting to close this question as a duplicate. – A.P. Jan 21 '16 at 01:14

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This seems unlikely, since there are plenty of bounded subsets of the real line having no maximum or minimum element. The natural numbers are well-ordered; this is induction. The reals are not.

ncmathsadist
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