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February 22nd, 2017, 05:48 AM   #1
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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 11:23 AM.
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February 22nd, 2017, 11:05 AM   #2
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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.
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February 27th, 2017, 09:37 AM   #3
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Quote:
Originally Posted by Magnitude View Post
Hi,

1) (lecture definition) every convex subset of the real numbers is an interval.

2) (book) every sequence of real numbers which is increasing and bounded above converges to a real number.

James
Wicki: metric space M is complete if every Cauchy sequence in m converges to a point in M.

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)
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