Completeness Properties of $\mathbb{R}$: Least Upper Bound Property, Monotone Convergence Theorem, Nested Intervals Theorem, Bolzano Weierstrass Theorem, Cauchy Criterion.
Archimedean Property: $\forall x\in \mathbb{R}\forall \epsilon >0\exists n\in \mathbb{N}:n\epsilon >x$
I can show that LUB implies the Archimedean Property but what about the rest of these properties? Please provide proofs (even hints) or counterexamples.
EDIT: It was shown by Isaac Solomon that the Bolzano-Weierstrass implies the Archimedean Property.