I consider here the Cauchy equivalent classes of Cauchy sequences which I assume relates to the the limits of the sequences. I assume further that the Cauchy sequences involved cannot be described by finite formulated formulas since they would be countable and cannot be in a one-to-one correlation to the real numbers. This is also supported by Wikipedia’s description of Cauchy sequences which gives the alternate description of a real number x as a limit of a series made up by “the successive truncations of the decimal expansion of x”.
The successive truncation definition is obviously also depending of uncountable definitions, so that there is a one-to-one correspondence between the signifier: the definition, and the signified: the real numbers. In essence, it looks very much like the definition of real numbers is done using real numbers. Isn’t that a problem?
No, the definition of real numbers via Cauchy sequences is not circular as you seem to suggest. It is true that the set of Cauchy sequences of rationals is not countable. However, we don't need it to be countable. We use one set of cardinality $c$ to define another, new, set of cardinality $c$. This is not circular.