Linear continua and the Real numbers

Simple question, probably a simple answer, but I'm stuck.

I'm trying to prove that the set of real numbers (in the usual order topology if you want specifics) is a linear continuum. I am having a problem with the least upper bound property. As I understand it, for the reals to have the lub property, any subset of the reals must have a lub. So what is the lub of the interval (1,+infinity)? It has no lub in the reals. But if it has no lub, then the set of real numbers doesn't have the lub property and thus is not a linear continuum! I've obviously gotten something wrong here, but I don't know what.

Any help would be appreciated!

Thanks.

Dan