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.
Best Answer
For Bolzano-Weierstrass $\to$ Archimedean Property: Take $\epsilon > 0$, and consider the sequence $\{n\epsilon: n \in \mathbb{N}\}$. If the Archimedean Property fails, then this sequence is bounded, so that it has a convergent subsequence. However, it is easy to see that this sequence does not have a convergent subsequence.
For Monotone Convergence Theorem $\to$ Archimedean Property: Taking the same sequence $\{n\epsilon: n \in \mathbb{N}\}$, it is easy to see that it is monotone. If the Archimedean Property fails, then this sequence is bounded, so by the MCT, it would have a finite limit, which is not the case.