Re: Replacement of Cardinality

Le 15/08/2024 à 19:01, Moebius a écrit :

Assume that there is an x e IR such that NUF(x) = 1. Let x0 e IR such that NUF(x0) = 1. This means that there is exactly one unit fraction u such that u < x0. Let's call this unit fraction u0. Then (by definition) there is a (actually exactly one) natural number n such that u0 = 1/n. Let n0 e IN such that u0 = 1/n0. But then (again by definition) 1/(n0 + 1) is an unit fraction which is smaller than u0 and hence smaller than x0. Hence NUF(x0) > 1. Contradiction!

We can reduce the interval (0, x) c [0, 1] such that x converges to 0.
Then the number of unit fractions diminishes. Finally there is none remaining. But never, for no interval (0, x), more than one unit fraction is lost. Therefore there is only one last unit fraction.
 Regards, WM