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

On 9/7/24 9:03 AM, WM wrote:

On 07.09.2024 14:37, Richard Damon wrote:

On 9/7/24 8:04 AM, WM wrote:

 

NUF(x) changes by 1 because a change by more at any x would count more different unit fractions 1/n,  1/m, 1/k, ... which are identical because they are the same x = 1/n = 1/m = 1/k = ... .
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And thus is always has a value of aleph_0 for all x > 0, since there is always aleph_0 unit fractions below and finite positive x value.

 Stop that nonsense. ℵo unit fractions cannot fit into every interval (0, x). The distance between two unit fractions already is larger. And if you claim that ℵo unit fractions exist, then you must accept that their distances do exist too.
 By the way this is independent of the existence of NUF.
 Regards, WM

Of course the can. Let N = floor(1/x)
Then we can put 1/(N+1), 1/(N+2), 1/(N+3), ... 1/(N+n), ... for all finite additions n, all aleph_0 of them into that interval.
Show me a value of x that doesn't hold.
Note, the distance between all of them exists, but the sum of the distances between of all the unit fractions is just 1, and the sum of the distances between all of the unit fractions smaller than x is less than x.
The concept of finite but unboundedly small seems beyound your minds ability to process.