1. Let (s n ) be a bounded sequence and let S denote the set of subsequential limits of (s n ). Prove that S is closed 2. Let A = {x ∈ Q: 0 ≤ x  → A. Letting s (n) = s n , find the set of...

1. Let (s_n) be a bounded sequence and let S denote the set of subsequential limits of (s_n). Prove that S is closed

2. Let A = {x ∈ Q: 0 ≤ x <> → A. Letting s (n) = s_n, find the set of subsequential limits of the sequence (s_n).

3. Let (sn) and (tn) be bounded sequences.

(a) Prove that lim sup (s_n
+ t_n) ≤ lim sup s_n
+ lim sup t_n.

(b) Find an example to show that equality may not hold in part (a).