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.
Topologically, extend a theorem to a dense subset of the "desired" set.
Algebraically, for example in vector spaces, use the bases and add elements.