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Unique infimum proof

  1. Mar 15, 2012 #1
    1. The problem statement, all variables and given/known data
    Let A be a subset of ℝ, now prove if inf(A) exists, then inf(A) is unique

    3. The attempt at a solution
    I am using an epsilon proof but i dont think i am going about it the right way, can someone nudge me in the right direction
     
  2. jcsd
  3. Mar 15, 2012 #2

    Ray Vickson

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    How you do it depends very much on what *definition* you are using for "inf". Different (but equivalent) definitions would need different types of proofs.

    RGV
     
  4. Mar 15, 2012 #3
    Usually, for something like this, it is best to start of saying "Let x and y be inf(S)" and then show that x and y are, in fact, the same. Now, showing that |x - y| < epsilon might be a good strategy, but there are probably more efficient methods. But, as RGV said, it depends on how inf is defined for you. If you have defined it as a greatest lower bound, the method I mentioned works very well.
     
  5. Mar 15, 2012 #4
    yes i think im going to revise it as x and y are infimums for A then x<y and y<x so x=y
     
  6. Mar 15, 2012 #5
    That's a contradiction the way you wrote it. Did you mean [itex]x \leq y, \ \text{and} \ y \leq x,\ \text{so} \ x=y[/itex]?
     
  7. Mar 15, 2012 #6
    yes that is what i meant and i did write it that way x≥y y≥x
     
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