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CHAPTER II [:] The Real Number System [...] 3. THE LEAST UPPER BOUND PROPERTY. "end quote
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LUB4: RATIONALS WITH A GIVEN DENOMINATOR NEAR A GIVEN REAL

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For any   x   and positive integer   N,   there is an integer   n   such that   x .
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See the added details following Rosenlicht's proof:

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To show this we merely have to apply LUB 3 to the number   Nx,   getting an integer   n,   such that   nNxn+1 .
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To get the desired result, we multiply all terms in   nNxn+1   by 1/N.   This is valid for the equality by F4. For the inequalities, we first use O7 to show that   1/N   is greater than zero. Then use O4 twice - once for each inequality.

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