Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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!

  2. jcsd
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?
Draft saved Draft deleted