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 s,, is in S . Suppose S is closed. Let (s, ) be a convergent sequence of points in S . = s. E S for all n e N There are two cases to consider. Let x = lim s, . if s. = z for some n E N , then x = lim sn e S if s, = z for all n E N , then (8n) must be a convergent sequence of points in S\\{x } that converges to I . By the statement proved in part one of question 5 above, I must be an accumulation point of S . Prove that a 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 (8, ) converges to I . S contains all its accumulation points by Theorem 3.4.17. r= lim s. e S Hence, in either case, lim on is in S