Would this argument be valid for proving A for all real numbers greater than or equal to a:
Prove that A is true at n = a.
Assume that A is true for all $ a\le n < k$. Prove that A is true at k.
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See Pete L. Clark's nice introduction to real induction, which may have been sparked by by this prior question Induction on real numbers. – Bill Dubuque Dec 14 '13 at 02:43
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He wrote a slightly newer version here. – Akiva Weinberger Feb 27 '15 at 05:40
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No, because your inductive step assumes that you know $A$ holds on some initial interval of the form $[a,k)$, but you have only shown that $A$ holds at a single initial point.
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Well if you show $A$ holds on the initial interval $[a,k)$, then show that $A$ holds on $[a,k)$ implies $A$ holds on $[a,k]$, you'll still only have shown that $A$ holds on $[a,k]$. I suggest you look at the links Dubuque posted to see how induction on the reals can work. – Dec 15 '13 at 00:23