Perhaps, due to our interest in things categorical, we can enjoy (instead of Cauchy sequence methods) to see the order of the (extended) real line as the [Dedekind-MacNeille_completion of the rationals]( Matthew has told us interesting things about it [before](

Hausdorff, on his part, in the book I mentioned [here](, [says]( that any total order, dense, and without \\( (\omega,\omega^*) \\) [gaps](, has embedded the real line. I don't have a handy reference for an isomorphism instead of an embedding ("everywhere dense" just means dense here).