
Real Analysis Real Analysis Math Forum 
 LinkBack  Thread Tools  Display Modes 
February 22nd, 2017, 04:48 AM  #1 
Newbie Joined: Jan 2016 From: United Kingdom Posts: 29 Thanks: 0  The Completeness Axiom Of the Reals
Hi, I've been taught the completeness axiom for the real numbers in my calculus lectures, but whilst reading an analysis book I discovered an axiom with the same name, but different in its presentation: (lecture definition) every convex subset of the real numbers is an interval. (book) every sequence of real numbers which is increasing and bounded above converges to a real number. For both, I can see there is a need to "fill in" the real number line. But is one implied by the other? Surely there can be only one such "Axiom"? What's the difference, if any? Regards, James Last edited by skipjack; February 22nd, 2017 at 10:23 AM. 
February 22nd, 2017, 10:05 AM  #2 
Senior Member Joined: Aug 2012 Posts: 1,428 Thanks: 352 
It's normal for there to be several equivalent characterizations of some property, and for different texts to use one as the definition and prove the others as theorems. Since the conditions are equivalent (meaning that if you assume either one you can prove the other) it makes no difference.

February 27th, 2017, 08:37 AM  #3  
Senior Member Joined: Mar 2015 From: New Jersey Posts: 1,081 Thanks: 87  Quote:
1) means there are no missing points on any line in S (Every Cauchy Sequence converges). 2) is a Cauchy sequence (No missing points on any line)  

Tags 
axiom, completeness, reals 
Thread Tools  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
Completeness Relation  GradualTint  Complex Analysis  0  May 24th, 2015 11:17 AM 
Is the completeness of the real number line equivalent to Dedekind's axiom?  RyanPowers  Real Analysis  7  September 25th, 2014 10:55 AM 
Completeness, consistency, and what we can know  v8archie  Math  3  April 7th, 2014 07:30 PM 
Countability and completeness  Huitzyl  Real Analysis  1  August 2nd, 2012 02:28 PM 
Gödel's Completeness Theorem  Hyperreal_Logic  Applied Math  0  December 31st, 2009 05:27 PM 