The "principle of mathematical induction" says that for a subset $S$ of $\omega$ (where $\omega$ is the set of all natural numbers), if $0 \in S$ and $n \in S \implies n^+ \in S$, then $S = \omega$.
The "principle of transfinite induction" says that if $X$ is a well-ordered set, $S \subset X$, and $s(x) \subset S \implies x \in S$ (where $s(x)$ is the set of all predecessors of $x$ in $X$), then $S = X$.
Page 67 of Naive Set Theory (Halmos) says that "[the principle of transfinite induction] when applied to $\omega$ is easily proved to be equivalent to the principle of mathematical induction".
However, I had trouble proving this (possibly because I had trouble formulating this "equivalence" precisely).
How can I formulate and prove the statement that "the principle of transfinite induction, when applied to the natural numbers, is equivalent to the principle of mathematical induction"?