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    Real Analysis : Convergent and Cauchy Sequences - Five Problems

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    ? For each of the follwing statements decide if it is true or false. Justify your answer by proving, or finding a couter-example.

    1) every bounded sequence of real numbers is convergent.
    2) Every convergent sequence is monotone.
    3) Every monotone and bounded sequence of real numbers is a cauchy sequence.

    ? For each of the following statements decide if it is true or false, justify your answer by proving or disproving the statement:

    1) a sequence of real numbers can have different subsequences which converge to different limits.
    2) A convergent sequence of rational numbers is a cauchy sequence.

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    https://brainmass.com/math/real-analysis/real-analysis-convergent-cauchy-sequences-28850

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    • For each of the follwing statements decide if it is true or false. Justify your answer by proving, or finding a couter-example.

    1) every bounded sequence of real numbers is convergent.
    False, a counterexample is :
    1,-1,1,-1,1,-1,....,(-1)^(n-1),...

    2) Every convergent sequence is monotone.
    False, a counterexample is

    It is convergent with limit 0 but ...

    Solution Summary

    Convergent and Cauchy Sequences are investigated. The solution is detailed and well presented.

    $2.49

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