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From: "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com>
Newsgroups: sci.math
Subject: Re: How many different unit fractions are lessorequal than all unit
 fractions? (infinitary)
Date: Wed, 30 Oct 2024 13:00:51 -0700
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On 10/30/2024 5:57 AM, FromTheRafters wrote:
> WM explained :
>> On 29.10.2024 11:02, FromTheRafters wrote:
>>> WM formulated the question :
>>>> On 28.10.2024 21:51, FromTheRafters wrote:
>>>>> WM explained :
>>>>>> On 28.10.2024 15:07, FromTheRafters wrote:
>>>>>>> WM was thinking very hard :
>>>>>>
>>>>>>>> The function NUF(x) = Number of Unit Fractions between 0 and x 
>>>>>>>> starts with 0 at 0.
>>>>>>>
>>>>>>> Followed by a discontinuity.
>>>>>>
>>>>>> No. ∀n ∈ ℕ: 1/n - 1/(n+1) > 0 holds everywhere.
>>>>>
>>>>> Not for n equal to zero.
>>>>
>>>> I said ∀n ∈ ℕ. 0 is no natural number. NUF(0) = 0.
>>>
>>> That is why your function is a step function with a discontinuity.
> 
> I meant 'step-wise' function.
> 
>> NUF(0) = 0, NUF(1) = ℵo. Therefore NUF must grow but cannot grow by 
>> more than 1 at any point x of the real axis.
> 
> The number of unit fractions less than x is always aleph_zero for 
> positive x. Nothing about growing them in some sort of unit fraction 
> backyard garden involved.

lol!