Question

Suppose that every Cauchy sequence of X has a convergent subsequence in X. Show that X is complete.

Answer #1

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Prove that every bounded sequence has a convergent
subsequence.

Prove that X is totally bounded if every sequence of X has a
convergent subsequence. Please directly prove it without using any
theorem on totally boundedness.

Show that sequence {sn} converges if it is monotone
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Exercise 2.4.5: Suppose that a Cauchy sequence {xn} is such that
for every M ∈ N, there exists a k ≥ M and an n ≥ M such that xk
< 0 and xn > 0. Using simply the definition of a Cauchy
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If ( |x_n| ) is Cauchy, then (x_n) is Cauchy.

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subsequence.

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Let X = (xn) be a sequence in R^p which is convergent
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show that the sequence sin(n+2)/n is Cauchy

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