I'm trying to solve problem 4.4.8 of Hrbacek & Jech's Introduction to Set Theory. The problem reads in part,
Let $(A,<)$ be linearly ordered. Define $\prec$ on $\operatorname{Seq}(A)$ by: $\langle a_0, \dots, a_{m-1} \rangle \prec \langle b_0, \dots, b_{n-1} \rangle$ if and only if there is $k < n$ such that $a_i = b_i$ for all $i < k$ and either $a_k < b_k$ or $a_k$ is undefined (i.e. $k = m < n$)... If $(A, <)$ is well-ordered, $(\operatorname{Seq}(A), \prec)$ is also well-ordered.
This looks to me like the lexicographic ordering of dictionary words, where a finite sequence that extends a shorter sequence comes after in the ordering. But then the sequence of strings $ab > aab > aaab > \cdots$ is an infinite decreasing sequence of words in $\operatorname{Seq}(A)$, which I think proves that $\operatorname{Seq}(A)$ cannot be a well ordering.
I must be making a simple mistake here, but in two days I haven't managed to find it! Can someone point out where I am going wrong?