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Sum of infs?

  1. Feb 14, 2012 #1
    Hi, I'm just wondering this:
    If A and B are subsets of the real numbers, is
    inf A + inf B = inf(A+B)?

    I've proved that
    inf A + inf B <or= inf(A+B)?

    I've tried the rest of the proof but it won't work. Is it even true?
     
  2. jcsd
  3. Feb 14, 2012 #2

    mathman

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    Let AI = inf(A), BI = inf(B) for every ε > 0, there exists a (in A) and b (in B) such that
    a < AI + ε and b < BI + ε, a + b < AI + BI + 2ε, inf(A+B) < AI + BI + 2ε .

    Since ε is arbitrarily small, inf(A+B) ≤ inf(A) + inf(B).
     
  4. Feb 15, 2012 #3
    Thanks. I got it in the end but your way is much prettier.
     
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