I never got a clear answer to this question in college. What happens in an induction proof if the assumption is wrong? For example, suppose we try to prove that $n^5$ > n! for n >= $2$ so we start out by stating that when n = $2$ it is true and in our proof, we assume that $n^5$ > n! and then try to show that it is also true for the n+1 case? Will something show up to invalidate the proof? If so where and how? Will it show us that the inequality will fail when n is $8$ or more or will it tell us it will fail but not inform us exactly where?
This might be an overly simple example so my more generic point is someone very sophisticated in math might be able to "hide" a flaw in their proof such that only another sophisticated math person might catch it, but to a not so obvious inequality like this one.
To me, it seems ok in an induction proof to show a specific case that works such as for n = $2$ and then try to prove it is true for all n, but to me is it NOT ok to assume that $n^5$ > n! for some arbitrary n because that is what we are trying to prove.
Obviously $n^5$ > n! will fail when n is $8$ or larger ($32,768$ is not larger than $40,320$).
Best Answer
If what you're assuming is wrong, then either the induction step or the initial condition is simply not possible to prove.