[ajuc]

It's not circular, because you aren't using the assumption A in the final proof, you are just using the implication A(N) => A(N+1) in the final proof.

the final proof is:

    A(1) is true
    AND
    A(N) implies A(N+1)
    means that
    A(N) is true for all N >= 1

the assumption that A is true for N was just used as a condition in the implication

If A(N) is false in general - you would realize that despite assuming it - because the implication or the base case couldn't be proven.