Graduate Real Analysis research based question- Completion of the real numbers

Question

I am currently a graduate student taking a real analysis independent study class. This is my first time taking real analysis, as I did not take it as an undergraduate. I am working on a research paper on the Completion of the Real Numbers. I am focusing on defining the real numbers using Cauchy sequences and Dedekind cuts. I am trying my best to research, however since this is my first time really taking a real analysis class I am a bit overwhelmed at the enormous amount of sources that I currently have. I am trying to understand the main sequence of proving that the real numbers are complete using Cauchy sequences. I understand that we want to show that each real number can be defined as a Cauchy sequence of rational numbers. However, since there are different Cauchy sequences of rational numbers that could potentially converge to the same real number. Therefore, it is necessary to introduce an equivalence class on C (the class of all Cauchy sequences of real numbers.) So we can let R denote the collection of all equivalence classes in C. What happens from there in order to complete the real numbers? I know that we must somehow show that the least upper bound of every subset of R is in R. But what are the necessary theorems that must be proven before showing this? In other words, can anyone provide me with a general outline of the steps necessary to show this?