I know that similar questions have been asked before, but I can't seem to find something that really justifies the Cauchy construction of the reals. One question that seemed to have been asked is how can Cauchy sequences exist without the real numbers. That's fine. You can use rationals. But more fundamentally, metrics are defined as real values functions. In fact, if you defined metrics as being rational valued functions, you run into the problem that there concept of metric falls apart for the reals.
Dedekind cuts avoid all of this and honestly seem like the "correct" way to define the reals. Unless there is a way to define a metric without them.