Gaps... ;^)

Between zero and any positive non-zero x there is a unit fraction small enough to fit in the gap. The x can even be a real that is not a unit fraction.
Between x and any y that is different than it (x), there will be a unit fraction to fit into the gap. infinitely many.... :^)
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.
Any thoughts? Did I miss something? Thanks.