I have to prove that for any $n\in\mathbb{N}$, $n\neq0$, $\exists k\in\mathbb{N}|n=S(k),$ where $S$ is the successor function.
I have to use the formal construction of Natural numbers.
I believe it can be proved by induction but the instruction is to prove it using a well-ordered set $(n,\in_{n})$, and $B\subseteq\mathbb{N}|B\neq0$ and its maximum.
$S(k)$ is the successor of $k\in\mathbb{N}$. $S(k)=k\cup\lbrace k\rbrace$