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There are many equivalent versions of completeness in the real number system:
i) LUB/supremum property
ii) Monotone Convergence property
iii) Nested Interval property
iv) Bolzano Weierstrass property
v) Cauchy Criterion property

I've been able to prove: (i)$\implies$(ii)$\implies$(iii)$\implies$(iv)$\implies$(v)
I need help with
a) (v)$\implies$(i)
b) (iii)$\implies$(i)


P.S. In proving (v)$\implies$(i), we use the construction of 2 sequences by using mid-points. I am having problem with showing that they are Cauchy sequences

  • 0
    Can you explain the cauchy criterion?2011-10-25
  • 0
    @Henrique, every Cauchy sequence converges.2011-10-25
  • 0
    If you know that (v) $\implies$ (i), you know that (iii) $\implies$ (i), since $\implies$ is transitive.2011-10-25
  • 1
    Are you looking to *explicitly* prove (iii) $\implies$ (i) (perhaps as an exercise), without going through (iv) and (v)?2011-10-25
  • 1
    It is economical to do a "round trip" of implications but it is very instructive to consider direct implications. Some work out nicely in a natural way; others seem to need to go through an auxiliary property. (I'm not sure this is the case here, but I've seen this happen often.)2011-10-25
  • 2
    Why does (iii) imply (i)? How do you know the naturals are not bounded?2011-10-25
  • 1
    As scineram has suggested, some of these equivalences may need the [Archimedean property](http://en.wikipedia.org/wiki/Archimedean_property), which is actually a consequence of (i).2011-10-27

2 Answers 2