Kurs:Mathematik für Anwender (Osnabrück 2019-2020)/Teil I/Repetitorium/8/Bounded and Cauchy/Studentenfrage/Antwort
In fact the three following statements for an archimedean ordered field are equivalent
- Every bounded above nonempty subset has a supremum (i.e. a least upper bound).
- Every bounded above increasing sequence converges.
- Every Cauchy sequence converges (i.e. the field is complete).
Maybe that is what you mean. But the implications are not as direct as it seems from your assertion as for example most Cauchy sequences are not monotonic.
However if you view this equivalence as one of these being true before the others you should ask yourself why the first assertion is true. In fact one common model for the real numbers (which we don't do in this lecture) is constructed specifically so that Cauchy sequences are convergent. You can also construct the real numbers in different ways where other equivalent properties are true by construction. Then you can deduce all the equivalent completeness properties from the one you constructed with. Thus if you want to think about this further it makes more sense to think about how to construct the real numbers.
For the sake of this lecture and for applications, however, it is best to view the completeness of the real numbers just as an Axiom. As such it is true without proof, because it is one of the defining features of the real numbers. A construction of the real numbers just gives proof that such a thing as the real numbers actually exists.