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I Need help understanding text from 'Calculus' by Apostol

  1. May 1, 2016 #1
    Sorry if I'm posting in the wrong place. Since the difficulty I have is with the text of the book, and not the exercises, I posted here.

    In proving theorem I.2, how is theorem I.1 used to assert that 'there is at most one such x'? The first image below gives the background text. The text I have trouble with is highlighted in the second image below.



    Thank you.
  2. jcsd
  3. May 1, 2016 #2


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    Assume that x and y satisfy theorem 1.2.

    That means that:
    a + x = b
    a + y = b

    What can you deduce from that about a + x and a + y?
  4. May 1, 2016 #3
    But I can't assume that. Theorem I.2 is the one I'm trying to prove.
  5. May 1, 2016 #4


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    You are questioning the part of theorem 1.2 that 'there is at most one such x'.
    You prove that by assuming that there are two, and then (hopefully), conclude that they must be equal.
    That in itself indeed doesn't prove that "there exists" an x satisfying theorem 1.2, but that if one exists, it is unique.
    Do you see how that uniqueness follows from theorem 1.1?
  6. May 1, 2016 #5
    Oh, I see now.

    So I have come up to the point saying a + x = b.

    But I do not know yet if x is unique. So I assume there is a another y such that a + y = b.

    Then, a + x = a + y

    From theorem I.1, x = y.

    Therefore x is unique.

    Thank you!
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