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How to write a mathematically rigorous definition of completeness

  1. Oct 10, 2009 #1
    I know that the definition of completeness is that a set contains the limits of rational numbers.

    and I know the definition of convergence is that for all e>0 there exists N such that for n>=N |xn - x| < e where x is the limit of the sequence.

    how to combine the two?
    thanks in advance
  2. jcsd
  3. Oct 10, 2009 #2
    I know the definition in terms of metric spaces, so maybe this is what youre looking for....

    A metric space (E,d) is called complete if every Cauchy sequence in E converges in E.

    of course the definition for cauchy sequence is a sequence given any e>0
    there is a positive integer N such that d(pm,pn) < e whenever n,m >N
  4. Oct 10, 2009 #3


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    There are, in fact, six properties of the real numbers that are equivalent to "completeness".

    1. The least upper bound property (every non-empty set having an upper bound has a least upper bound) and its "twin" the greatest lower bound property.

    2. Monotone convergence (every increasing sequence having an upper bound converges and its "twin" that every decreasing sequence having a lower bound converges.)

    3. The Cauchy Criterion (every Cauchy sequence converges)

    4. The Bolzano-Weierstrass property (every bounded sequence contains a convergent subsequence.)

    5. Every closed and bounded set is compact.

    6. The set of all real numbers, with the usual metric, is a connected set.

    Given any one of those you can prove the other five.
    Last edited by a moderator: Oct 11, 2009
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