Show n-1 is a natural number

Question

Show that $n \in N, n>0 \Rightarrow n-1 \in N$. We can use the basic axioms and we also know that $N$ is the smallest possible set $A$, where $A$ meets the following criteria: 1) $0 \in A$ 2) $\forall x \in R: x \in A \Rightarrow x+1 \in A$. My reasoning is the following. We can choose $A= \ldots \cup \{-1\} \cup \{0\} \cup \{1\} \cup \ldots$ to satisfy both conditions. We notice that $x \in A \Rightarrow x-1 \in A$. Since $N \subset A$, we have $n-1 \in N$. Is this correct? Additionally, do we have an induction proof for this?