 If we understand the theory of the reals prior to the cuts, we can say that each cut is of the form \(L = (-\infty , r) \cap \mathbf{Q}, R = [r, \infty) \cap \mathbf{Q}\), where \(\mathbf{Q}\) is the set of all rationals and \(r\) is a unique real number uniquely determining and uniquely determined by the cut. It is obvious that each real number \(r\) uniquely determines a cut in this way (but we need to show that there are no other cuts). Given an arbitrary cut \((L, R)\), we propose that \(r\) will be the least upper bound of \(L\). The Least Upper Bound Axiom of the usual theory of the reals tells us that \(L\) has a least upper bound \((L\) is nonempty and any element of \(R\) (which is also nonempty) is an upper bound of \(L\), so \(L\) has a least upper bound). Because \(L\) has no greatest element, its least upper bound \(r\) cannot belong to \(L\). Any rational number less than \(r\) is easily shown to belong to \(L\) and any rational number greater than or equal to \(r\) is easily shown to belong to \(R\), so we see that the cut we chose arbitrarily (and so any cut) is of the form \(L = (-\infty , r) \cap \mathbf{Q}, R = [r, \infty) \cap \mathbf{Q}\).