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Proof l.u.b of A.

  1. Oct 12, 2011 #1
    1. The problem statement, all variables and given/known data
    A= (4+X)/X WHEN x≥1
    Find the least upper bound and greatest lower bound of the following sets.for a given ε>0,
    Find a number in the set that exceeds l.u.b. A - ε and a number in the set that is smaller than
    g.l.b.A + ε


    2. Relevant equations

    does this make sense? is it a right way to attack this question?



    3. The attempt at a solution

    x≥1 → 1 ≥1/x → 0<1/x≤ 1
    is we multiply by 4. →0 < (1/x)4 ≤4
    then add one → 1 < (4/x)+1 ≤ 5.
    thus 1 is G.l.b of A AND 5 l.u.b of A.

    Find a number in the set that exceeds l.u.b. A - ε ....

    LET 5 is the upper bound of A. (given)
    suppose K is also the upper bound of A. → 5 ≤ K
    Suppose this is not the case.AND K < 5.
    5-K > 0
    ε = 5-K > 0
    K =5 -ε SO THAT there is a number X(ε) E A thus x(ε) > 5-ε = K
     
  2. jcsd
  3. Oct 13, 2011 #2

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    Yes, 5 is the lub (in fact, it is the maximum value) of A and 1 is the glb.

    But I'm not sure what you are doing with:
    Well, it's not given- you had to find that 5 is the least upper bound.

     
  4. Oct 13, 2011 #3
    would you use the same strategy for G.L.B TOO...WOULDN'T THE NEGATIVE CHANGE THE EQUALITY
     
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