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    Convergent Sequence of Rationals

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    Show that, for every real number y, there is a sequence of rational numbers which converges to y.

    © BrainMass Inc. brainmass.com February 24, 2021, 2:32 pm ad1c9bdddf
    https://brainmass.com/math/real-analysis/convergent-sequence-rationals-26945

    Solution Preview

    There are two cases to consider:

    Case i. y is rational

    For every n >= 1, let a_n = y - (1/n). Then a_n is rational for every n >= 1, since (a) both y and 1/n are rational and (b) the difference of two rational numbers is rational. Clearly, the sequence {a_n} converges to y.

    Example: y = 3.

    The sequence of rational ...

    Solution Summary

    A detailed proof of the following statement is given: For every real number y, there is a sequence of rational numbers which converges to y.

    $2.19

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