Example 3 (APMO) Find all nonempty finite sets $S$ of positive integers such that if $m, n \in$ $S$, then $\frac{m+n}{(m, n)} \in \mathbf{S}, (m, n$ do not have to be distinct).
Analysis We start from a special case, first knowing that $2 \in S$, and then proceed step by step to explore other possible elements in the set $S$, finding that the set may contain only the element 2. How can we express this concisely?

Solution Let $m=n$, then $2 \in S$. Since $S$ is a non-empty finite set, if $S$ contains an odd number, then $\frac{k+2}{(k, 2)}=k+2 \in S$, and so on, $k+4, k+6, \cdots$ are all in $S$, which contradicts the fact that $S$ is a finite set. Therefore, all elements in $S$ are even. If there are other even numbers besides 2, let the smallest number other than 2 be $k(k>2)$, then $\frac{k+2}{(k, 2)}=\frac{k}{2}+1 \in S$, and $2<\frac{k}{2}+1<k$. From the previous discussion, $\frac{k}{2}+1$ should be even, which contradicts the assumption that $k$ is the smallest number other than 2. Therefore, $S=\{2\}$.
Comment The use of the extreme principle makes the expression concise.