anonymous
  • anonymous
m,n are natural numbers, S:={1/n-1/m}, find infS, supS.
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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chestercat
  • chestercat
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amistre64
  • amistre64
can 1/n = 1/m ?
amistre64
  • amistre64
i assume your text defines natural numbers as positive integers
anonymous
  • anonymous
yes 1/n can equal 1/n and yes N is positive >0 integers.

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amistre64
  • amistre64
the biggest value we can get is a 1 then; and the smallest approaches 0 so my thoughts are 0-0 = 0 1-0 = 1 0-1 = -1 1-1 = 0
anonymous
  • anonymous
I want to use the Archimedean property to ensure these extremum, but I am new to the property and an not clear about the corollaries that infer n (sub y) -1<=y<=n (suby)... I believe there will be more to a proof of this conclusion... you know?
amistre64
  • amistre64
im not familiar with the Archimedean property to be able to comment on its usages
anonymous
  • anonymous
consider that we can make \(1/n\) arbitrarily close to \(0\) and the maximum value of \(1/m\) is strictly \(1\), so our infimum or greatest lower bound is \(0-1=-1\) for our supremum or least upper bound, consider we can make \(1/n=1\) and then make \(1/m\) arbitrarily close to zero, so we get \(1-0=1\)
anonymous
  • anonymous
the archimedean property comes in here by guaranteeing that getting arbitrarily close to \(1\) from below or to \(-1\) from above precludes there being any other candidate supremum/infimum, since that would suggest that there is some 'infinitesimal' number

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