How can I show the set $\{n \mid n = 1 \text{ or } n \in \mathbb{N} \text{ and } n - 1 \in \mathbb{N}\}$ is inductive? A set is inductive if it contains the element $1$, and if $x$ is in the set, then $x + 1$ is also in the set.
By the set definition, we know $1$ is in the set. But how can I prove the other condition? seems like it's going in the other direction.
If you assume that $x$ is in your set and want to prove that $x+1$ is in your set as well, you need to show that $x+1 \in \mathbb{N}$ and $x + 1 - 1 = x \in \mathbb{N}$. The latter condition is given by the inductive hypothesis. What is left to prove is that $x + 1 \in \mathbb{N}$.