Mark each statement True or False. Justify each answer. (a) Every sequence has a convergent subsequence. (b) The set of subsequential limits of a bounded sequence is always nonempty. (c) (s n )...

Mark each statement True or False. Justify each answer.

(a) Every sequence has a convergent subsequence.

(b) The set of subsequential limits of a bounded sequence is always nonempty.

(c) (s_n) converges to s iff lim inf s_n
= lim sup s_n
= s.

(d) Let (s_n) be a bounded sequence and let m = lim sup s_n. Then for every ε > 0 there are infinitely many terms in the sequence greater than m − ε .

(e) If (s_n) is unbounded above, then lim inf s_n
= lim sup s_n
= +
.