Can induction be used for $n \leq 0 $

Question

Example question

Prove that $ exp(n) = e^{n} \space \space \forall \space n \in \mathbb{Z} $

First I prove by induction for $ n \geq 0 $ and then I do the same for $ n \leq 0 $

Is this allowed ?

Answer 1

Yes. If you have a proposition $P(n)$ for $n<0$, consider $Q(n)=P(-n)$, for $n>0$.
If you can prove $Q$ by induction, then you have proved $P$.