An example of a total order (that is NOT a well-order) of the Natural numbers

Question

I need an example of a total ordering of the Natural numbers, that is not a well-ordering. So the classic "less than or equal to" doesn't work in this case since it is well-ordered.

I've been wracking my brain for hours but I can't just conjure up a relation. Is there a relatively simple one I'm forgetting?

Answer 1

How about:$$\ldots,7,5,3,1,0,2,4,6,\ldots$$

Or even simpler, just turn $\Bbb N$ around!

Added: To make that last idea more precise, define a relation $\preceq$ on $\Bbb N$ by $m\preceq n$ if and only if $n\le m$. Then check that $\preceq$ is a linear order on $\Bbb N$ that is not a well-order. (The last is clear, since for any $n\in\Bbb N$, $n+1\preceq n$, and therefore $\Bbb N$ has no $\preceq$-least element.)

Answer 2

What about reverse order? i.e. $n\prec m$ if $n>m$.