# Equivalent formulations of completeness

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