Re: How many different unit fractions are lessorequal than all unit fractions?

On 9/9/2024 3:57 PM, Chris M. Thomasson wrote:

On 9/9/2024 3:50 PM, Chris M. Thomasson wrote:

On 9/9/2024 8:01 AM, WM wrote:

On 09.09.2024 13:47, Richard Damon wrote:

On 9/9/24 6:31 AM, WM wrote:

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And all gaps are occupied by the unit fractions. Hence every gap is too small.

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But there is always room at the bottom, where the gaps keep getting smaller

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They all are present from the start. I simply choose a gap that is too small to contain ℵo unit fractions.

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Wtf are you talking about? The gaps keep getting smaller:
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1/1 - 1/2
1/2 - 1/3
1/3 - 1/4
1/4 - 1/5
1/5 - 1/6
1/6 - 1/7
...
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Forever.... They still get arbitrarily close to zero. Don't try to mix and match reals with the unit fractions willy nilly. All unit fractions are reals, not all reals are unit fractions. See?
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How can a claim a set of values less then a SPECIFIC NUMBER if I don't have the number?

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I claim that all unit fractions are existing as real points of the real line. Therefore there is a first one. NUF(x) cannot increase without passing 1 when real points are counted. Your ℵo points cannot exist without including 1, 2, 3 first points.
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Regards, WM
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 Between zero and any positive non-zero x there is a unit fraction small enough to fit in the gap. The x can even be a real that is not a unit fraction.
 Between x and any y that is different than it, there will be a unit fraction to fit into the gap. infinitely many.... :^)
 Say the gap is abs(x - y) where x and y can be real. If they are different (aka abs(x - y) does not equal zero), then there are infinitely many unit fractions that sit between them.

The difference between any two points can be (is already) normalized where 1/2 is half way there. Two points wrt they are not identical to each other.