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Homework Help: Analysis Equality

  1. Sep 7, 2010 #1
    1. The problem statement, all variables and given/known data

    Prove that the real numbers a and b are equal if and only if for each positive real e, the absolute value of a-b satisfies abs(a-b)<e

    2. Relevant equations

    The main one I am thinking about is the fact that if a<=b and b<=a then a=b, also the whole sigma thing might mean the archimedean property might come into effect (if r and s are positive ration numbers then there exists a positive integer N such that rN<s).

    3. The attempt at a solution
    I am tried a proof by contradiction.... but generally it falls apart or requires too much non-rigorous work :-/

    Any ideas/hints/help?
  2. jcsd
  3. Sep 7, 2010 #2
    okay; so I have a new idea after another half hour of work, i sorta feel though the last step is flawed;

    proof by contradiction

    a=b iff abs(a-b)>e
    for some e>0

    abs(a-b)>e means either
    a-b>e or -(a-b)>e
    if a-b>e then a>e+b
    and if -(a-b)>e then b>e+a

    however from these two points it is "obvious" that either a>b or b>a, but is there some theorem which actually proofs that?
  4. Sep 7, 2010 #3
    First of all, the iff means you have to show that both implications are true.

    The forward implication is that if a = b, then for every e > 0, |a-b| < e. This should be evident.

    The reverse implication is that if for every e > 0, |a-b| < e, then a = b. To do a proof by contradiction, you have to negate this last statement first. The negation of 'if p then q' for statements p and q is 'p and not q'. Assume the 'p and not q' condition and see if you can find a contradiction.
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