Ok, so i did this proof and I'm unsure about it?
I wnat to prove that has a convergent subsequence.
Since is a bounded sequence of real numbers, it must have at least one accumulation point. Let A be that accum pt. By definition, A is an accum pt of iff each neighborhood contains points of which are not A. Let be arbitrary. So, we can let elemnet of (A- ,A+ ). As ,(A- ,A+ ) accumulates at A. Therefore, A is an accum pt of .
I'm afraid I wasn't specific enough and should have included the case for finitely many distinct terms.