if I want to proof something for a restricted finite number of elements, meaning the following:
Imagine that I have a theorem that is somehow similar to the following: For each element in $\mathbb{A}^{n \times n}$, there is a statement that depends on a parameter $i\in {0,...,n}$ true, let me say $A^i$ has i zero rows(of course this is nonsense, but I want to ask whether the kind of proof is appropriate)!
Imagine that I would make in induction over n and then an induction over i, of course the induction over n is a valid method in order to try to proof this, but what is about this induction over i would it be also correct?
e.g. I could say in the induction over i: Hey look, for i=0 our Matrix $A$ is always of $rank(A)=n$ therefore it has no zero row
now the induction step could be $A^{k-1}$, where k-1 is less than n has k-1 zero rows, therefore $ A^{k-1}A$ has k zero rows. Of couse all these statements are nonsense, but my question is, if you proof something for a somehow finite but not fixed set is induction then necessary or is there another method way to do this?