Could someone please help me correct this proof

. ( + ) whenever (Sn) is a convergent sequence of points in S , it follows that lim on is in S = S is closed Suppose (Sn) is a convergent sequence of points in S and lim Sn E S. Let x 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 c . Sn E S and Sn * * = lim sn for all n E N. Since lim Sn E S and Sn E S , Since x E S' and & E S , = S' C S, which means that S contains all its accumulation points. By Theorem 3.4.17, S is closed