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How does one generally extend a theorem proved over the integers to the real numbers and beyond e.g. induction proofs, De Moivre's Theorem?

I am aware that to extend a theorem proved over $\mathbb{N}$ to $\mathbb{Q}$ requires substitutions of the kind $n=\frac{p}{q}, p,q \in \mathbb{N}$ and showing that it still holds. But is there a similarly general principle for extending a proof of such sorts to $\mathbb{R}$, and if so, what is it? Does a similar proof strategy exist for extending a proof to the complex field, or others?

I AM NOT asking about how to prove a theorem by induction over the real numbers, I am asking about the general principles of extending a theorem.

James Harrison
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  • Many theorems involve arithmetic functions which cannot be defined over the real numbers. I believe that if a theorem starts with "Suppose $n$ is rational" is just because the theorem holds only for rationals. I believe you are asking something too general to be answered. – Konstantinos Gaitanas Oct 02 '14 at 08:00
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    While I admit that it is general, I did not cite any theorems that started with "suppose n is rational", I cited De Moivre's Theorem specifically because it could be proved over the integers and possibly extended into reals and complex numbers by a relatively general principle that can be universally applied (since it is in fact true over $\mathbb{C}$). IMOMath in this page cites a "standard method for extending to $\mathbb{R}$" but I am unsure how this is universally applicable: http://www.imomath.com/index.php?options=341&lmm=0 – James Harrison Oct 02 '14 at 08:11

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Topologically, extend a theorem to a dense subset of the "desired" set.

Algebraically, for example in vector spaces, use the bases and add elements.

Haha
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