Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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:

    Let

    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

    mathman

    User Avatar
    Science Advisor

    A consists of all integers < x, so it is bounded from above.
     
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook




Loading...
Similar Threads for Proof Integer Parts Date
B Proof of a limit rule Dec 19, 2017
B Proof of quotient rule using Leibniz differentials Jun 10, 2017
B Don't follow one small step in proof Jun 10, 2017
Addition property of integration intervals proof Feb 26, 2015