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Question involving graph

  1. Feb 3, 2013 #1
    1. The problem statement, all variables and given/known data

    Supopse that "a" and "b" are two numbers and that a < b+ε for all ε>0. Show that a ≤ b

    2. Relevant equations

    None given for problem

    3. The attempt at a solution

    See attachment of my jpeg for my attempt. I'm not sure, but i feel like it satisfies the statement
     

    Attached Files:

  2. jcsd
  3. Feb 3, 2013 #2

    haruspex

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    The graph does not constitute an argument. I would not even consider drawing a graph for this question (unless instructed to). Or are you just checking you have understood the question?
     
  4. Feb 3, 2013 #3
    For problems like these, can I use real numbers to try and figure it out?

    Assuming a = 5 and b + e = 6. These two values would work because 5 < 6. Since e > 0, it has to be a positive number. To make things easy, we'll pick the smallest positive integer greater than zero: 1. If e becomes equal to 1, then I can say that b is equal to 5. Now I can say that a = b, which satisfies the first part of the second equation.

    "b" can be greater than "a" if I pick any number I want "b" to equal, so long as it is greater than "a" by a substantial amount.

    Algebraically, I could write that

    a < b when e > (b-a), and a = b when e = (b-a) ?
     
  5. Feb 3, 2013 #4

    haruspex

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    This is not a question about integers, I'm sure. It's about reals. And no argument based on specific numbers could be considered a general proof.
    Try reductio ad absurdum: assume a is not <= b and see if you can demonstrate this violates "a < b+ε for all ε>0".
     
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