Dedekind cuts ?
That is what your attachment suggests you are working with. I have a small complaint about the handout. It should made it clear that in all cases
From that web page in the link that .
Note that in the example, we can define but there is no greatest or least element in one or the the other set. Hence a gap.
Between any two rational there is a third rational number, density.
But you have shown that there is no integer between so cannot be dense. By the same logic it cannot have gaps.
Does that help?