Something occurred to me about the method of proof of induction and i don't know if there is something in this?

Is it possible to use induction to prove results other than ones involving natural numbers?

So instead of, check n=1, assume true for n=k, prove n=k+1 works etc, can you do something like, check n=1/2, assume true for n=1/k, prove it works for n=1/(k+1) , hence true for n=1/3, 1/4 etc.

Can this method be extended even further to prove everything except results involving irrationals?

This may be gibberish...