I've been having a bit of a a think about this question but still haven't come up with a particuarly good answer yet.
So what's wrong with the following proof that x^n=1 for all non negative intergers n, and all non zero real x. Let P(n) be the proposition that x^n=1.
The base case x^0=1 (so that's fine)
Inductive hypothesis: suppose n greater than equal to 0, and that P(m) true for all m less than or equal to n. (i.e we're assuming everything up to P(n) is true. Then x^n=x^(n-1)=1 based on our assumption.
So p(n+1) is true. So by complete induction P(n) is true.
Thanks in advance