Let $S \subseteq \mathbb Z$, prove that if $0∈S$, and $k∈S$ implies $k+1∈S$ and $k-1∈S$, then $S=\mathbb Z$

Question

Let $S \subseteq \mathbb Z$, prove that if $0∈S$, and $k∈S$ implies $k+1∈S$ and $k-1∈S$, then $S=\mathbb Z$

My intended approach is to convert this question into another form where I can do 2 mathematical inductions. One for $P(k)$ implies $P(k+1)$ and another for $P(k)$ implies $P(k-1)$.

However, since mathematical induction only work to prove $P(n)$, where $n∈ \mathbb N$, I am having trouble establishing the premises for these 2 inductive statements.

So, any possible proposed solution?

Answer 1

You can prove by induction that for $n\in\Bbb N$, if $n\in S$ and $-n\in S$, then both $\pm (n+1)\in S$.