Well Ordering implies Induction Proof doubt

Question

I’m trying to understand the proof for the fact that that the Principle of Well-Ordering implies the Principle of Mathematical Induction; that is, if S ⊂ N such that 1 ∈ S and n + 1 ∈ S whenever n ∈ S, then S = N.

In the above proof, how did they conclude that $k$ is in the set $S$. Thanks

Answer 1

By construction, the smallest element of $S'$ has the form $k+1$ and so $k<k+1$ lies in $S$ as $\Bbb N_0 = S\cup S'$.

Answer 2

The minimum element of $S^{'}$ is $k + 1$, by the set difference $k \in S$. The key is that the union of $S$ and $S^{'}$ is $\mathbb N$.