Could someone please check my work

2. Prove that a set S is closed iff, whenever (Sn) is a convergent sequence of points in S , it follows that lim sn is in S . . ( = ) S is closed = whenever (Sn) is a convergent sequence of points in S , it follows that lim sn is in S . Suppose S is closed. Let (Sn) be a convergent sequence of points in S . # 8n E S for all n E N There are two cases to consider. Let x = lim Sn . if 8n = x for some n E N , then x = lim on E S if 8n # x for all n E N , then (Sn) must be a convergent sequence of points in S\\{x} that converges to x . # By the statement proved in part one of question 5 above, ax must be an accumulation point of S . Prove that x is an accumulation point of a set S iff there exists a sequence (Sn ) of points in # Since S is closed, S\\{x} such that (Sn) converges to x . # S contains all its accumulation points by Theorem 3.4. 17. = x = lim 8n E S Hence, in either case, lim Sn is in S