Proof of the Equivalence of Completeness Properties - MathOverflow [closed] most recent 30 from http://mathoverflow.net 2013-06-20T00:08:15Z http://mathoverflow.net/feeds/question/88531 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/88531/proof-of-the-equivalence-of-completeness-properties Proof of the Equivalence of Completeness Properties Stephen 2012-02-15T16:55:04Z 2012-02-15T16:55:04Z <p>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.</p> <p>I can show 1→2 and 1→3→4→5→1 All I need to prove their equivalances is 2→3</p> <p>I therefore need the proof of the Monotone Convergence Theorem using Nested intervals Theorem</p> <p>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.</p> <p>MCT:If $a_{n}$ is a monotone and bounded sequence of real numbers then $a_{n}$ converges.</p>