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**jsmith90210** 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?

Thanks