I have a general question about strong induction:
Assuming that the base case is 0, if I let my inductive hypothesis be that for all 0 <= k < n some statement is true, and if I prove that that statement is true for n, does this mean that I've proved it for all natural numbers?
Some examples do something a little differently: Assuming that for all 0 <= k <= n is true which includes n, they prove it's true for n + 1..are these two equivalent? Or is only the second method valid? Thanks.