Re: Gaps... ;^)

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.