Induction over real number

Question

I have to prove a property $P(x)$ hold for $\forall x: x \in (0,1]$. I also have a property $F\big(\frac{x}2\big)=F(x)+1$ which is key to prove $P(x)$. If I prove following steps:

$P(x-\epsilon)$ holds for $\forall \epsilon$,where $\frac{x}{2}<x-\epsilon<x$
$P\big(\frac{x}{2}\big)$ holds
$\forall x \in (0,1],\space \space \exists \frac{x}{2} \in (0,1]$.

It this a correct way to prove using induction over real number?

I guess that's $\forall \epsilon \dots$ ? – DanielV Feb 23 '15 at 08:23 — DanielV, Feb 23 '15 at 08:23
yes thanks a lot..i have update the question – Tom Feb 23 '15 at 09:34 — Tom, Feb 23 '15 at 09:34

score 4 · Accepted Answer · edited Apr 13 '17 at 12:20

4

You're not the only one asking this question. Pete Clark has given it a lot of thought and gives a good answer in this discussion.

edited Apr 13 '17 at 12:20

Community

1

answered Feb 23 '15 at 07:09

Mathemagician1234

17,966

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Slightly relevant (uses transfinite induction to prove a property on $(0,1)$) – Akiva Weinberger Feb 23 '15 at 07:57

Induction over real number

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