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Proof of Integer Parts of Real numbers

  1. Jul 25, 2010 #1
    I am struggling to understand the proof for integer parts of real numbers. I have used to mean less than or equal to because I could not work out how to type it in. I need to show that:

    ∃ unique n ∈ Z s.t. nx<n+1

    The proof given is the following:


    A={k∈Z : kx}

    This is a non-empty subset of R that is bounded above. Let α = sup A. There is an n ∈ A such that n > a - 1/2. n>α−1. Then nx and,since n+1>α+1>α, n+1̸∈A. Hence,n+1>x.

    In particular I don't understand how A is bounded above, because I thought A = [k,∞) which has no upper bound. Where have I gone wrong?
  2. jcsd
  3. Jul 25, 2010 #2


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    Science Advisor

    A consists of all integers < x, so it is bounded from above.
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