I found the following statement in Munkres' Topology:
Theorem 4.2 (Strong induction principle). Let $A$ be a set of positive integers. Suppose that for each positive integer $n$, the statement $S_n \subset A$ [here $S_n = \{1, 2, \dots, n\}$] implies the statement $n \in A$. Then $A = \mathbb{Z}_+$.
Now, I think I understand strong induction. But what I don't understand here is, doesn't $S_n \subset A$ always imply $n \in A$? It's pretty much by definition of a subset, that if $\{1,2,\dots,n\} \subset A$ then $n \in A$. Did the author mean "Suppose that for each positive integer $n$ the statement $S_n \subset A$ is true"?
If you check the previous page, you should see that $S_n$ is the set of positive integers that are less than $n$. $S_n$ here denotes the section of $\Bbb Z_+$ by $n$. See also the more general definition immediately before Lemma 10.2.