Let < be a lexicographic order on $^{\omega}2$ or in other words given distinct functions $f,g$ from $\omega$ to 2, let $f<g$ if and only if $f(n)=0$ and $g(n)=1$, where $n$ is the lease $m<\in\omega$ such that $f(m)\neq g(m)$. How can we see ($^{\omega}2$,<) is not well-order. 



($^{\omega}2$,<) is not well-order.
Let $(f\_n)$ be the point that maps $\{0,\dots,n\}$ to $0$ and the rest to $1$. 
Then $f\_{n+1} < f\_n$ for all $n$, showing an infinite descending sequence.


