1. (a) Suppose that lim s n = 0. If (t n ) is a bounded sequence, prove that lim (s n t n ) = 0. (b) Show by an example that the boundedness of (t n ) is a necessary condition in part (a). 2. Suppose...

1. (a) Suppose that lim s_n
= 0. If (t_n) is a bounded sequence, prove that lim (s_nt_n) = 0.

(b) Show by an example that the boundedness of (t_n) is a necessary condition in part (a).

2. Suppose that (a_n), (b_n), and (c_n) are sequences such that a_n
≤ b_n
≤ c_n
for all n ∈ N and such that lim a_n
= lim c_n
= b. Prove that lim b_n
= b.

3. Suppose that lim s_n
= s, with s > 0. Prove that there exists N ∈
 such that s_n
> 0 for all n ≥ N.