Ok, so i did this proof and I'm unsure about it?

I wnat to prove that

has a convergent subsequence.

Proof:

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.