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

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- Apr 8th 2007, 02:33 PMslowcurv99Cauchy subsequence and a limit
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

- Apr 8th 2007, 02:44 PMThePerfectHacker

A)Every bounded sequence has a convergent subsequence. That sequence is a Cauchy sequence.

B)No. s_n=n.

Quote:

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