1)
prove or give a counterexample:
A) every bounded sequence has a Cauchy subsequence
B) every monotone sequence has a bounded subsequence
2)
Suppose that x>1. Prove that Lim x^(1/n) = 1
A)Every bounded sequence has a convergent subsequence. That sequence is a Cauchy sequence.
B)No. s_n=n.
1<= x <= n for sufficiently large n.2)
Suppose that x>1. Prove that Lim x^(1/n) = 1
Then,
1<=x^{1/n} <= n^{1/n}
Squeeze theorem.
Use famous fact that,
n^{1/n} --> 1