I was recently studying the construction of real numbers. I read that the sum of 2 reals using the left Dedekind sets was the set of sum of all the rational numbers contained within those two sets.
What I am not able to understand is how the sum of the rational numbers of those two sets contains all the rational numbers less than the real number associated in the sum. In other words, how can I be sure that there is no rational number greater than the greatest sum of the rationals in the two sets but at the same time lesser than the resulting real number ? [ I am aware that I should not be thinking about the greatest rational in the set, but I am inclined to think about the greatest number when taking the sum as it should represent the largest of the rationals in the resulting set ]
Any help would be greatly appreciated. Thanks in advance.