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

On 9/28/2024 3:04 AM, FromTheRafters wrote:

Chris M. Thomasson was thinking very hard :

On 9/27/2024 3:13 PM, FromTheRafters wrote:

WM expressed precisely :

On 25.09.2024 19:22, Richard Damon wrote:

On 9/25/24 11:14 AM, WM wrote:

NUF increases. At no point it can increase by more than 1.

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Why not?

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Because there is a finite gap between two unit fractions.

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There are no gaps in the set of unit fractions.

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He might be thinking

 But I rather doubt it.
 

of the gap between say 1/1 and 1/2, its there. Think of the "granularity" of _strict_ unit fractions for a moment.

 There are gaps with respect to a domain other than the unit fractions. There are gaps in the set of primes with respect to the set of naturals, that is, these gaps are filled with composite numbers in the naturals.
 There are gaps in Q with respect to R which are filled with irrationals thus making R complete (meaning no gaps).
 

Then think of trying to fill the gap with unit fractions breaking the strict rule for sure:
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1/2 + ((1/16*4 + 1/8*3 + 1/4 + 1/8) - 1/2) = 1
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;^)

 Here you've gone from unit fractions to all of Q+ I think. Yes, there are gaps in Unit Fractions with respect to Q+ yet filling them with other elements of Q+ does not 'complete' the set with respect to Q or R.
 Here, I'm thinking of gaps as meaning places where a cauchy sequence does not converge to a member of the set but somehow would fit between them in an extended domain which included that element.

Wrt to R, there are no gaps, right? It's infinitely dense.
However, _strictly_ confining ourselves to the strict _order_ of unit fractions, well, there are gaps:
1/1 - 1/2 = 1/2 gap
No we know that there are infinitely many unit fractions smaller than the gap for sure. However, using a strict order and strictly unit fractions,
1/1, 1/2, 1/3, 1/4, ...
Well, we can't put anything between 1/1 and 1/2 for this would _break_ the rules wrt the word, strict...
Humm... Sound okay to you? Thanks.