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Math Help - analysis

  1. #1
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    analysis

    Any ideas on this problem? Thanks!

    Let a be a positive number. Prove that for each real number x there is an integer n such that na is less than or equal to x is less than (n+1)a.
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  2. #2
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    Quote Originally Posted by friday616 View Post
    Any ideas on this problem? Thanks!

    Let a be a positive number. Prove that for each real number x there is an integer n such that na is less than or equal to x is less than (n+1)a.
    You know that there is an integer n such that na>x - by the Archimedean property. Consider the set S=\{ n | na \leq x \}. This set must be bounded by what we have just set above. Let N = \max\{ S\}. Then we have that Na \leq x, but by construction N+1\not \in S since it is larger than the maximum and so (N+1)a > x.
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