Could someone please help me fix this proof

. ( + ) whenever (Sn) is a convergent sequence of points in S , it follows that lim Sn is in S = S is closed Suppose (Sn) is a convergent sequence of points in S and lim Sn E S. Let a E S' so that x is an accumulation point, then by the statement proved in part one of question five above, there exists a sequence (Sn) of points in S\\{x} such that (Sn) converges to x . Sn E S and Sn * * = lim Sn for all n E N. Since lim Sn E S and Sn E S , =&ES Since x E S' and a E S , = S' C S, which means that S contains all its accumulation points. By Theorem 3.4.17, S is closed