1. Let (s n ) and (t n ) be bounded sequences. (a) Prove that lim inf s n + lim inf t n ≤ lim inf (s n + t n ). (b) Find an example to show that equality may not hold in part (a). 2. Let (sn) be a...

1. Let (s_n) and (t_n) be bounded sequences.

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

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

2. Let (sn) be a bounded sequence.

(a) Prove that lim sup s_n
= lim N →
 sup {s_n: n > N}.

(b) Prove that lim inf s_n
= lim N →
 inf {s_n: n > N}.