The completeness properties are 1)The least upper bound property, 2)The Nested Intervals Theorem, 3)The Monotone Convergence Theorem, 4)The Bolzano Weierstrass, 5) The convergence of every Cauchy sequence.
I can show 1→2 and 1→3→4→5→1 All I need to prove their equivalances is 2→3
I therefore need the proof of the Monotone Convergence Theorem using Nested intervals Theorem
The theorems: NIT:$I_{n}=\left [ a_{n},b_{n} \right ]$ and $I_{1}\supseteq I_{2}\supseteq I_{3}\supseteq...$ then $ \bigcap_{n=1}^{\infty}I_{n}\neq \varnothing$ In addition if $b_{n}-a_{n}\rightarrow 0$ as $n \to \infty$ then $\bigcap_{n=1}^{\infty}I_{n}$ consists of a single point.
MCT:If $a_{n}$ is a monotone and bounded sequence of real numbers then $a_{n}$ converges.
