In current teaching materials, when using Dedekind cuts to construct real numbers, the definition of a Dedekind cut is always involved in defining addition and multiplication. In this paper, as it is done in many current textbooks, Dedekind cuts are used to construct the set of real numbers. Then the order in it is defined, and the least-upper-bound property, as well as the density properties of rational cuts in the set of real numbers, is established. Later, when defining addition and multiplication and establishing relative properties, the definition of cut will not be used again. This makes the process simpler and easier to understand.