Let A be a nonempty subset of the Reals bounded above. Suppose sup A exists and that sup A is not an element of A. Show that A contains a countably infinite subset. In particular, A is infinite.
I'm reasonably sure that the easiest way to prove this is using an epsilon argument, but I don't really know where to start on here.
Here's my scrambled thoughts:
Let a = sup A
x be an element of A
x <= a
For any epsilon > 0 there is a-epsilon < x < a
how do a I show that the interval from a-epsilon to x is countably infinite?