Proof by induction on an uncountable interval

Asked Oct 04 '18 at 23:17

Active Oct 05 '18 at 01:44

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I'm a little bit confused as to whether or not I can use proof by induction on an uncountable interval. For example, I'm trying to prove that $$(x+y)^n \leq x^n + y^n$$ on the interval $0 \leq n \leq 1$.

Primarily, I'm stuck on what my base case would be. Just n = 0?

asked Oct 04 '18 at 23:17

bco

4

Induction on an uncountable interval is a different logical beast than ordinary induction. For instance, the interval $0 \leq n \leq 1$ does not satisfy the Well-Ordering Principle. So induction is probably not the intended answer here. – Matthew Leingang Oct 04 '18 at 23:23
Possible duplicate of Prove, by induction, $x\ne 1\implies x+1\ne2$ for all $x$ in $[0,1]$ – ℋolo Oct 04 '18 at 23:38
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The "duplicate" is about the general case of "induction over an interval" and more, it will help the confusion but not specifically about this case – ℋolo Oct 04 '18 at 23:40
Maybe you could try to do induction for a countable subset of the interval, and see how the two sides behave when you put $n$ tends to an irrational... To do that you might want the subset to be dense possibly. I have not tried myself, so the above is pure guessing. – AdditIdent Oct 05 '18 at 01:24
Yes, the base case is $n=0$. – Fabio Somenzi Oct 05 '18 at 01:59

Proof by induction on an uncountable interval

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