Lexicographic Order on Products of Well-Ordered Sets

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:
 * For a given $$n \in \N: n > 0$$, $$\preccurlyeq$$ is a well-ordering on the set $$T_n$$ of all ordered $n$-tuples of $$S$$;
 * $$\preccurlyeq$$ is not a well-ordering on the set $$T$$ of all ordered tuples of $$S$$.

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

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

Proof for Fixed n
Consider $$T_n$$ where $$n \in \N: n > 0$$.

It is clear that $$\left({T_1, \preccurlyeq}\right)$$ is order isomorphic to $$\left({S, \preceq}\right)$$.

Thus as $$\preceq$$ is a well-ordering on $$S$$, $$\preccurlyeq$$ is a well-ordering on $$T_1$$.

Now, let us assume that $$\preccurlyeq$$ is a well-ordering on $$T_k$$ for some $$k \in \N: k \ge 1$$.

Let $$A$$ be a non-empty subset of $$T_{k+1}$$.

Let $$A_1$$ be the set of all of the first components of the ordered $n$-tuples that comprise $$A$$.

Since $$A_1$$ is a non-empty subset of $$S$$, and $$S$$ is itself well-ordered by $$\preceq$$, it follows that $$A_1$$ contains a minimal element $$x$$ by $$\preccurlyeq$$.

Let $$A_x$$ be the subset of $$A$$ in which the first component equals $$x$$.

We may consider $$A_x$$ to be a subset of $$T_k$$ where this first component $$x$$ has been suppressed.

But we assumed that $$T_k$$ is well-ordered by $$\preccurlyeq$$.

So $$A_x$$ contains a minimal element $$\left({x, x_2, x_3, \ldots, x_{k+1}}\right)$$ by $$\preccurlyeq$$.

This element $$\left({x, x_2, x_3, \ldots, x_{k+1}}\right)$$ is the minimal element of $$A$$ by $$\preccurlyeq$$.

Hence, by definition, $$T_{k+1}$$ is well-ordered by $$\preccurlyeq$$.

The result follows by induction.

Proof for Variable n
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 $$\preccurlyeq$$.