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Equivalent formulations of completeness

  1. Nov 17, 2013 #1
    So far, I have encountered the following formulations of completeness and was wondering whether they are all equivalent:

    1) Supremum principle: Every nonempty bounded subset of ℝ has a supremum.
    2) Infimum principle: Every nonempty bounded subset of ℝ has an infimum.
    3) Monotone sequence property: Every monotonic bounded sequence in ℝ converges to an element of ℝ
    4) Dedekind completeness: If S and T form a Dedekind cut of ℝ such that S < T, then either S has a largest element or T has a smallest element.
    5) Cauchy completeness: Every Cauchy sequence in a metric space converges to an element of that metric space.
    6) Nested interval property: The (infinite) intersection of every nested sequence of closed balls (in an arbitrary metric space) whose radii tend to zero is nonempty.

    I have proved equivalence of 1, 2 and 4. I have also proved that 1 implies 3. Is it true that 3 implies 1? Clearly, 5 and 6 are defined for arbitrary metric spaces, but do they become equivalent statements to the others if we regard ℝ as a metric space with the standard Euclidean metric?

    Thanks for clarifications. Do not state proofs. I just want to know whether they're equivalent. I will do proofs myself, thanks!

    BiP
     
  2. jcsd
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