Re: Does the number of nines increase?

On 7/9/2024 2:34 PM, Moebius wrote:

Am 09.07.2024 um 22:07 schrieb Chris M. Thomasson:
 > On 7/9/2024 5:11 AM, WM wrote:
 >> Le 09/07/2024 à 03:04, Moebius a écrit :
 >>> Am 07.07.2024 um 22:24 schrieb WM:
 >>>> Le 05/07/2024 à 15:54, Moebius a écrit :
 >>>>> Am 05.07.2024 um 09:39 schrieb WM:
 >>>>>  > Le 04/07/2024 à 17:25, Moebius a écrit :
 >>>>>  >> Am 04.07.2024 um 17:16 schrieb WM:
 >>>>>  >>
 >>>>>  >>> die Verteilung der ersten Stammbrüche
 >>>>>  >>
 >>>>>  >> es gibt keine "ersten Stammbrüche". Zu _jedem_ Stammbruch gibt es
 >>>>>  >> (abzählbar) unendlich viele kleinere Stammbrüche.
 >>>>>  >
 >>>>>  > Wrong,
 >>>>>
 >>>>> Nope.
 >>>>
 >>>> Insufficient argument.
 >>
 >>> Hinweis: Wenn s ein Stammbruch ist, dann ist 1/(1/s + 1) ein
 >>> Stammbruch, der kleiner ist als s.
 >>
 >> Not true for the smallest unit fraction.
  > There is NO smallest unit fraction. Dark or not. :^)
 Indeed!
 Theorem: There is no smallest unit fraction.
 Proof: If s is a unit fraction, then there's (by definition) a natural number n such that s = 1/n. Then n + 1 is a natural number to (by one of the peano axioms and the definition of +) and n + 1 > n (by the definition of >). Hence 1/(n + 1) < 1/n (by the rules of the field Q). And hence 1/(1/s + 1) < s (again by the rules of the field Q). Moreover 1/(1/s + 1) is a unit fraction (again by definition). Hence There is a unit fraction, namely 1/(1/s + 1), which is smaller than s.
 If we define:
         s is a smallest unit fraction =df s smaller than all other unit fractions
 Then the theorem we just proved implies that THERE IS NO "smallest unit fraction".

No smallest unit fraction because any unit fraction, say this one:
1/(huge_natural_number) has a smaller one at:
1/((huge_natural_number) + 1) which is smaller... ;^)

 HENCE in a mathematical context we would not be allowed to talk about "the smallest unit fraction" (as Mückenfuck does) simply because there IS NO such entity.
 You see:
 The round square is round an square.
The triange with 4 corners has 4 corners.

lol!! You made me laugh. A triangle with 4 vertices is an interesting one. lol. Thanks. :^)