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From: Moebius <invalid@example.invalid>
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Subject: Re: Gaps... ;^)
Date: Tue, 10 Sep 2024 02:28:44 +0200
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Am 10.09.2024 um 00:59 schrieb Chris M. Thomasson:

> Between zero and any positive x there is a unit fraction small 
> enough to fit in the ["]gap["].

Right. This follows from the so called "Archimedean property" of the 
reals. From this property we get:

For all x e IR, x > 0, there is an n e IN such that 1/n < x.

See: https://en.wikipedia.org/wiki/Archimedean_property

Of course, from this we get that there are infinitely many unit 
fractions smaller than x, say, 1/n, 1/(n + 1), 1/(n + 2), 1/(n + 3), ...

We can even refer to such unit fraction "in terms of x":

All of the following (infinitely many) unit fractions are smaller than 
x: 1/ceil(1/x + 1), 1/ceil(1/x + 2),  1/ceil(1/x + 3), ...

> Between x and any y that is different than it (x), there will be a unit 
> fraction to fit into the gap. infinitely many.... :^)

Nope. There is no unit fraction (strictly) between, say, 1/2 and 1/1.

In other words, there is no unit fraction u such that 1/2 < u < 1/1.

> Say the gap is abs(x - y) where x and y can be real. If they are 
> different (aka abs(x - y) does not equal zero), then there are 
> infinitely many unit fractions that sit between them.

Nope. See counter example above.

> Any thoughts? Did I miss something? Thanks.

Yes. It works for any (0, x) where x e IR, x > 0.

But it does not work "in general" for (x, y) where x,y e IR, x,y > 0 and 
x < y (and hence abs(x - y) > 0).

If you'd consider _rational numbers_ (or fractions) instead of unit 
fractions, your intuition would be right, though.