Proving a theorem for all real numbers using mathematical induction

Question

For the sake of this question, let us consider the De Moivre theorem. As of now, I am able to prove $(\cos(\theta)+i\sin(\theta))^{n} = \cos(n\theta)+i\sin(n\theta)$ for all integers(positive and negative).

Using the fact that the theorem holds good for integers, if I try to prove the theorem for $\frac{a+b}{2}$, where $a$ and $b$ are integers, will this prove the theorem for all real numbers?

Note: The intent of this question is NOT to prove the theorem for all real numbers, but to discuss whether proving a theorem for $\frac{a+b}{2}$ where $a$ and $b$ are integers is a valid method to prove for all real numbers

Answer 1

If this were a valid proof technique, you could use it to prove that all real numbers are rational: clearly all integers are rational, and if $\frac pq$ and $\frac rs$ are rational then so is $$ \frac{\frac pq + \frac rs}2 = \frac{ps + rq}{2qs}. $$ Therefore this is not a valid proof technique for proving something for all real numbers.

Answer 2

No it would not work as you suggested.

If you could prove the theorem for example for all rational numbers (more generally: any dense subset of the reals), then you could conclude that it holds for all real numbers by a continuity argument (the expressions occuring in the formula you gave as an example define continuous functions).

The problem you have is that expressions of the form $\frac{a+b}{2}$ where $a$ and $b$ are integers do not form a dense subset of the reals, so you cannot conclude that the formular holds for all real numbers.

Edit: Maybe I was a bit unclear: Of course the conclusion from the rationals to the reals is only possible if we want to prove a formular which depends continuously on the variable over which we are inducing. If that's not the case (as for example in Mees de Vries' answer) then this does not work. But as the OP explicitly gave an example where we actually have a formula depending continuously on the variable we are inducing I thought that this was actually the point he was interested in.