READING: II §3 LUB4
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II §3 LUB3
begin quote ita"
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
For any x and positive integer N, there is an integer n such that x .
See the added details following Rosenlicht's proof:
To show this we merely have to apply LUB 3 to the number Nx, getting an integer n, such that nNxn+1 .
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.
on to §3: LUB 5