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Real Analysis

  1. Nov 2, 2008 #1
    The problem statement, all variables and given/known data

    Let S be a set of positive real numbers with an infimum c > 0 and let the set T = {[tex]\frac{1}{t}[/tex] : t [tex]\in[/tex] S}.

    Show that T has a supremum and what is it's value.

    The attempt at a solution

    Ok, so the value must be [tex]\frac{1}{c}[/tex].

    But I'm unsure how to start proving that T must have a supremum. Any starting hints would be great :) thanks
     
  2. jcsd
  3. Nov 2, 2008 #2
    1/c is absolutely an upper bound, thus you need to prove it is the supremum, which means, for any e>0, you can find a 1/s such that 1/s > 1/c-e. find such s based on the fact that c is the infimum of S.
     
  4. Nov 3, 2008 #3
    Okay I think I have something but I'm unsure whether it's right.

    for [tex]\frac{1}{c}[/tex] = sup T it must meet the following two criteria.

    1) [tex]\frac{1}{c}[/tex] is an upper bound such that [tex]\frac{1}{c}[/tex] [tex]\geq[/tex] [tex]\frac{1}{t}[/tex] [tex]\forall[/tex] t [tex]\in[/tex] S

    2) [tex]\forall[/tex] e > 0, [tex]\exists[/tex] x [tex]\in[/tex] A with [tex]\frac{1}{c+e}[/tex]< [tex]\frac{1}{t}[/tex] [tex]\leq[/tex] [tex]\frac{1}{c}[/tex]

    So my attempted proof follows that we can argue by contradiction.

    supposing [tex]\frac{1}{c}[/tex] satisfies 1 and 2.

    So 1) [tex]\Rightarrow[/tex] [tex]\frac{1}{c}[/tex] is an upper bound

    Assume that [tex]\frac{1}{c'}[/tex] = sup T so [tex]\frac{1}{c'}[/tex] < [tex]\frac{1}{c}[/tex].

    From this I get for some e > 0 [tex]\frac{1}{c'}[/tex] = [tex]\frac{1}{c+e}[/tex].

    2) [tex]\Rightarrow[/tex] [tex]\exists[/tex] [tex]\frac{1}{t}[/tex] [tex]\in[/tex] T with [tex]\frac{1}{c+e}[/tex]< [tex]\frac{1}{t}[/tex] [tex]\leq[/tex] [tex]\frac{1}{c}[/tex] [tex]\Rightarrow[/tex] [tex]\frac{1}{c'}[/tex]< [tex]\frac{1}{t}[/tex] [tex]\leq[/tex] [tex]\frac{1}{c}[/tex] which contadicts the fact that c' = sup T

    Think I've made a bit of a mess of it as I'm trying to base it off an example thats kind of similar in my notes
     
  5. Nov 3, 2008 #4
    what is the x in your 2) criteria?
    What is the A in your 2) criteria?
    By the definition, you 2) criteria should be:
    for any e>0, there exists a t in S such that 1/t > 1/c - e.
    If you think your criteria is equavalent to this one, you have to show us. ( I havn't checked it)

    If you want to argue by contradiction, you made something wrong.
    You already assume that 1/c satisfies 1 and 2, which implies that 1/c is the least upper bound. you assume again that 1/d should be the least upper bound, which is less than 1/c. From these two assupmtions alone, you already get a contradiction
     
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