If we have a set $A$ that is defined as a non empty set in rational numbers. Then we use it's $\sup(A)$ to define real numbers as it is not always in the rationals.
My question is, does this mean that if we take $\sup(A)-\epsilon$,
$\epsilon$ being infinitely small, does that mean that that number is a rational and we can express it in $\frac{m}{n}$. Why wouldn't this be true if we have a proof for incompleteness of rationals that states that we can always find a $\beta^²$ that is between let's say the number $2$ and $\alpha^2$. So we can find a rational between any rational and irrational number as far as I understand. How "big" must be $\epsilon$ for us to move from the supremum to a rational number.
If my assumption in the second paragraph is already partially correct, does there exist an irrational number that is infinitely close to another irrational.
I am an undergraduate so that is why my question might be a little too basic.