A let xn be a bounded sequence prove that if limnrarrinfin

Question

a) Let {x_n} be a bounded sequence. Prove that if limn→∞ sup |x_n| = 0, then limn→∞ x_n exists and equals 0.

(b)  Prove that a bounded sequence that does not converge always has at least two subsequences that converge to different limits.

(c)  Find the limit inferior and limit superior of the sequence {a_n} if a_n = ⌊sin n⌋ for all n ∈ N.

(d)  Find the set of all subsequential limits for the sequence {x_n} if for all n ∈ N