I’ve noticed it’s conventional in most literature to state the base case as a lower bound in an inductive hypothesis. So for example if the base case is P(1) then in the inductive hypothesis P(k) it’s usually stated that k is a positive integer greater than or equal to 1.
But is it even necessary to state this? The goal of induction is to show that $P(k) \implies P(k+1)$ and use the base case to prove it for every integer onwards. To show this one must only show that $P(k+1)$ holds if $P(k)$ is assumed to be true so bounds don’t seem to matter if there isn’t a case where the first part is true but the second isn’t. (Of course you still have to prove the base case)
Is there any reason induction is usually written with bounds then?
More specifically, I'm asking if the highlighted part is actually necessary:
So far, I've seen cases of texts on induction including bounds and others not including restrictions at all.