Right... Dave pointed out my silly mistake. I was thinking of ordering in the order you read, that is things to the top come first and then you worry about left to right, whereas the way i've defined lexigraphic ordering it goes the other way. Dave's fix is just replacing x everywhere with y.

He's also asked: why do the left and right limits exist at dicontinuous points? Well the left limit is increasing and bounded above, and the right limit is decreasing and bounded below. That bounded monotonic sequences have limits is one of the many equivalent axioms of the reals...

Ok, nerdiness over... But Alec did say how much he was going to miss hearing us babble about math...