# Induction over the rationals?

Is there a variant form of induction to prove something about the rationals as opposed to just the natural numbers?

You could start by proving it for the open interval (0, 1) by showing that for an arbitrary integer m, m < n, $$P(\frac{m}{n}) \Rightarrow P(\frac{m}{n+1})$$, for all natural numbers n, and then extend the domain to all positive rationals.

Is this even plausible?

## Answers and Replies

You could start by proving it for the open interval (0, 1) by showing that for an arbitrary integer m, m < n, $$P(\frac{m}{n}) \Rightarrow P(\frac{m}{n+1})$$, for all natural numbers n, and then extend the domain to all positive rationals.

Certainly, try the dictionary or spiral ordering of rationals. For that matter, induction is applicable to any countable set.The practical appicability is limited, though.

Is there a variant form of induction to prove something about the rationals as opposed to just the natural numbers?

You could start by proving it for the open interval (0, 1) by showing that for an arbitrary integer m, m < n, $$P(\frac{m}{n}) \Rightarrow P(\frac{m}{n+1})$$, for all natural numbers n, and then extend the domain to all positive rationals .

Is this even plausible?
I would try proving it for the open interval (0, 1) by showing that for an arbitrary integer m, m < n, $$P(\frac{m}{n}) \Rightarrow P(\frac{m+1}{n})$$, for all natural numbers n, and then extend the starter domain to all positive rationals m/n, m<n. Where n = 1 you have the standard induction process.

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