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## Proof of the Equivalence of Completeness Properties [closed]

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.

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Dear Stephen, please read the FAQ: this website is for research level math questions. You should try math.stackexchange.com . – Angelo Feb 15 2012 at 17:18
LEMMA: Any non-empty subset $B$ of $\mathbb{R}$ contains a decreasing sequence $b_n$ with the same set of minorants as $B$ (that is, for any $\alpha\in\mathbb{R}$ we have $\alpha\le b$ for all $b\in B$ if and only if $\alpha\le b_n$ for all $n\in \mathbb{N}$. Then, given your bounded and monotone (say, increasing) sequence $a_n$, apply the lemma to the set $B$ of majorants of {$a_n$}. – Pietro Majer Feb 15 2012 at 17:27
Thaks for your answers. How do I delete a question? – Stephen Feb 15 2012 at 17:30
You're welcome. I think the closed question will be deleted after some time. – Pietro Majer Feb 15 2012 at 18:55