So I have a general idea of what the proof should like (construct a sequence in S, since it is bounded the sequence has a convergent subsequence, and so the limit of that subsequence is an accumulation point). However, I am having a hard time construction such a sequence such that none of the terms of the subsequence are the limit point. How would I go about doing this. I considered using the supremum of S as the limit point but if the supremum is in S, there is no guarantee that every single term of a constructed sequence that converges to the supremum has terms in which none of the terms are equal to the supremum itself. Any help would be appreciated, thanks.