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Math Help - infinite decimal expansion

  1. #1
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    infinite decimal expansion

    How to I show that every infinite string of decimal digits corresponds to a unique real number \beta using the completeness axiom? I am suppose to somehow construct a bounded set with supremum \beta but do not know how..this problem seems to be way beyond me.
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by Kiwili49 View Post
    How to I show that every infinite string of decimal digits corresponds to a unique real number \beta using the completeness axiom? I am suppose to somehow construct a bounded set with supremum \beta but do not know how..this problem seems to be way beyond me.
    I don't understand the question.

    Are you trying to show that if \beta\in\mathbb{R} there is an "infinite" decimal expansion for \beta? If you're counting things like .1\overline{0} then you're answer lies in considering a sequence of rational numbers which converges to \beta

    If you are given an infinite decimal expansion \sum_{n=-N}^{\infty}\frac{a_n}{10^n} it clearly (I hope ) represents a real number and if \beta=\sum_{n=-N}^{\infty}\frac{a_n}{10^n},\gamma=\sum_{n=-N}^\infty \frac{a_n}{10^n} then (foregoing convergence issues for now) \beta-\gamma=\sum_{n=0}^{\infty}0=0

    I can't say much more from there.
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