Let $P(n)$ be a statement to be proven by induction, where $n \in \mathbb{N}$ (with possible exceptions). In the inductive step we conventionally assume a statement $P(k)$ to be true, to show that $P(k) \implies P(k+1)$. Do we have to define $k$ as being natural (or commensurate to the domain of $P(n)$), instead of it just being an integer? The whole point of the inductive step is to show that a relation holds between two consecutive integer numbers, regardless of whether there is a first number to start from, so in principle it should also be valid for negative numbers.
Likewise, would it also make sense to use induction 'in reverse', for negative numbers, that is: Prove $P(n)$ for some $n \lt 0$, then assume $P(k)$ to be true, where $k \in \mathbb{Z^-}$ (or just $\mathbb{Z}$ for that matter), then shown that $P(k) \implies P(k-1)$?
