Lexicographic Order on Products of Well-Ordered Sets

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

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

Then:
 * $(1): \quad$ For a given $n \in \N: n > 0$, $\preccurlyeq_l$ is a well-ordering on the set $T_n$ of all ordered $n$-tuples of $S$;
 * $(2): \quad \preccurlyeq_l$ is not a well-ordering on the set $T$ of all ordered tuples of $S$.

Proof
It is straightforward to show that $\preccurlyeq_l$ is a total ordering on both $T_n$ and $T$.

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