Problem: Prove that a real number s_0 is an accumulation point of a set S if and only if there exists some sequence {a_n} in S such that a_n =/= s_0 for every n in N and lim n-> INF a_n = s_0.

I understand the "<=" implication. I am struggling with the "=>" implication. The definition of accumulation point I am using is "A point s_0 is an accumulation point of a set S if for every epsilon > 0, there exists a number t in S such that 0 < |t - s_0| < epsilon."

So far I no that t is in (s_0 - e, s_0 + e) but I am having difficulty showing there is a sequence {an} in (s_0 - e, s_0 + e) s.t. lim is s_0. I think that since t is in S and {an} is a sequence of numbers in S, that t should be in {an}. So, I have been finding sequences such that an = t/n. I don't know if a_n lies in the interval after a certain point or if s_0 even converges.

I've seen proofs using epsilon = 1/n and stating there exists a sequence of numbers in (s_0 - 1/n, s_0 + 1/n) but no one explains why this is true. I'm missing that little piece (may be obvious to other people). What is the reason that lets us simply say "there exists a sequence of numbers {an} that lies in the interval (s_0 - 1/n, s_0 + 1/n)? I thought we need to show there was a sequence of numbers not simply say there is.

Anyone please help me bridge this gap in this proof? Thank you.