How to Prove a Nonempty Set of Real Numbers is Not Sequentially Compact?

[atm06001]

Suppose that S is a nonempty set of real numbers that is not Sequentially compact. Prove that either (i) there is an unbounded seqeunce in S or (ii) there is a sequence in S that converges to a point x₀ that is not in S.

I am having trouble with this it not being sequentially compact is screwing me up, I don't know how to prove it.

 

[morphism]

Well, there are two ways you can proceed:
(1) Suppose that S is bounded, and prove that there is a sequence in S that converges to something outside of S, or
(2) suppose that every convergent sequence in S has a limit in S, and prove that S cannot be bounded.

Have you tried either way? If so, what kind of problems did you run into?