Let $\{A_n\}_{n\in\Bbb N}$ be a collection of well-ordered sets. $X$ is defined by $X=\prod_{n\in\Bbb N}A_n$. Find an example such that $X$ with the lexicographic order is not well-ordered.
I know that if $X$ were defined by finite Cartesian product, it would necessarily be a well-ordered set with the lexicographic order. But I don't know how to find an example so it won't be well ordered. Thanks
On
Let $A=\{0,1\}$ with $0<1.$ For $k\in N$ let $x_k=(x_{k,n})_{n\in N}\in A^N, $ where $x_{k,n}=1$ if $n\geq k,$ and $x_{k,n}=0$ if $n<k. $ Then $x_k>x_{k+1}.$
Remark : On the set $S$ of all $y\in [0,1)$ such that $y$ has a base-$3$ representation without using the digit $2,$ the natural (arithmetic) order on $S$ co-incides with the lexicographic order on the sequences of digits (without using the digit $2$) of members of $S.$ And this is just the lex order on $\{0,1\}^N.$ Clearly $S\cap (0,1)$ is not empty and has no least member.
Consider the collection $\{ A_n\}_{n\in\mathbb{N}}$ where $\forall n\in\mathbb{N}$, $A_n=\mathbb{N}$ each with the usual well-order given by $\leq$.
We will construct a decreasing sequence of elements as follows, to begin, choose an element $(a_1,a_2,...)\in\prod_{n\in\mathbb{N}} A_n$ so that none of the $a_i=1$. Then construct a new element, $(a_1-1,a_2,...)$ which is also in the product, but occurs earlier in the lexicographic order. Next we construct $(a_1-1,a_2-1,...)$ and so forth.
Thus we have a sequence $(\alpha_k)_{k\in\mathbb{N}}$, where
$\alpha_k=(a_1-1,...,a_{k-1}-1,a_k,...)$.
This sequence satisfies $\alpha_k\leq\alpha_{k-1}$ for all $k\in\mathbb{N}$ and hence the set $\{\alpha_k:k\in\mathbb{N}\}$ does not have a least element, and thus the lexicographic order is not a well-order.