1. If (s n ) is a subsequence of (t n ) and (t n ) is a subsequence of (s n ), can we conclude that (s n ) = (t n )? Prove or give a counterexample. 2. Let (s n ) be a bounded sequence and suppose...

1. If (s_n) is a subsequence of (t_n) and (t_n) is a subsequence of (s_n), can we conclude that (s_n) = (t_n)? Prove or give a counterexample.

2. Let (s_n) be a bounded sequence and suppose that lim inf s_n
= lim sup s_n
= s. Prove that (s_n) is convergent and that lim s_n
= s.

3. Suppose that x > 1. Prove that lim x^1/n
= 1