Let xn be the following sequence: 1/2, 1/3, 2/3, 1/4, 2/4, 3/4, 1/5, 2/5, 3/5, 4/5, 1/6, ...

Find the set S of all of its subsequential limits. (Hint: Theorem 52: [A number z is a subsequential limit of a sequence (xn)n if and only if for every > 0, the set of indices

{n N : |xn z| < } is infinite] will be very useful here. Make sure to prove both that (a) every element of S is a subsequential limit, and (b) every number not in S cannot be a subsequential limit.)

(Hint 2: after you finished your work, check: Theorem 58 says that S should be a closed set. Is the