Infinite Lexicographic Order on Well-Ordered Sets is not Well-Ordering

Theorem
Let $S$ be a set which is well-ordered by $\preceq$.

Let $\preccurlyeq$ be the lexicographic order on the set $T$ of all ordered tuples of $S$.

Then $\preccurlyeq$ is not a well-ordering on $T$.

Proof
It is straightforward to show that $\preccurlyeq$ is a total ordering on $T$.

It remains to investigate the nature of $\preccurlyeq$ as to whether or not it is a well-ordering.

Consider a set $S = \left\{{a, b}\right\}$ such that $a \prec b$.

Then the set $\left\{{\left({b}\right), \left({a, b}\right), \left({a, a, b}\right), \left({a, a, a, b}\right), \ldots}\right\}$ has no minimal element by $\preccurlyeq$.

Thus $T$ is not well-ordered by $\preccurlyeq$.