Exercise 11, § 2

Let I be a well-ordered set and let $(\mathbf{E}_\iota)_{\iota \in \mathbf{I}}$ be a family of ordered sets, each of which contains at least two distinct comparable elements. Then the lexicographic product of the $\mathbf{E}_\iota$ is well-ordered if and only if each of the $\mathbf{E}_\iota$ is well-ordered and I is finite (if I is infinite, construct a strictly decreasing infinite sequence in the lexicographic product of the $\mathbf{E}_\iota$). \*