20210426, 20:44  #12  
Aug 2005
Seattle, WA
1782_{10} Posts 
Quote:


20210426, 21:15  #13  
"Alexander"
Nov 2008
The Alamo City
3·11·23 Posts 
Quote:


20210427, 08:39  #14 
Dec 2012
The Netherlands
17×103 Posts 
In that case, it's time for our 2nd important general principle of the thread:
given any property, always consider whether the set of all elements with that property is just a subset or a substructure (subgroup, subring, subfield or whatever). Here: all you have to do is show that the set of rational numbers that have the Archimedean property form a subfield. 
20210429, 15:35  #15 
Feb 2017
Nowhere
2^{4}·313 Posts 
The Archimedean property is that if x and y are in an ordered field, x > 0 and y > 0, there is a positive integer n such that n*x > y.
Assuming the ordered field is the field of rational numbers with the usual ordering, I note that if x and y are positive rational numbers, there is a positive integer M such that M*x and M*y are both positive integers. Then taking n = M*y + 1, we have n*(M*x) >= n*1 = M*y + 1 > M*y, so that n*x > y. If we ignore the ordering, and use instead a "nonArchimedean valuation" (padic valuation), anything dependent on ordering (like "upper bound" or "least upper bound," and therefore "Dedekind completeness") goes out the window. Luckily, "Cauchy completeness" (every Cauchy sequence in the field has a limit in the field) can still be used to embed the padic rationals (and their extensions) into fields that are (Cauchy) complete WRT a nonArchimedean valuation. Last fiddled with by Dr Sardonicus on 20210429 at 15:35 Reason: xifgin posty 
Thread Tools  
Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
How do I prove (a+b)^2=a^2+2ab+b^2 from axioms?  drkirkby  Homework Help  10  20210422 15:41 
something I think is true but can't prove  wildrabbitt  Math  5  20190430 22:45 
So how must we be able to prove the following?  George M  Miscellaneous Math  5  20180102 11:11 
I have to prove everything  PawnProver44  Miscellaneous Math  40  20160319 07:33 
Is this to prove/already known?  MatWurS530113  Math  4  20070627 05:35 