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Let $a : \mathbb{N} \rightarrow \mathbb{Q}$ be a Cauchy sequence of rationals. Then is it correct to say that

$$\lim_{n \rightarrow \infty} a_n = \{x \in \mathbb{Q} : \exists n \in \mathbb{N} : \forall m \in \mathbb{N} : m > n \rightarrow x < a_m\}$$

where the right side is interpreted as a lower Dedekind cut? That is, can we use this expression to convert any Cauchy sequence to a Dedekind cut that represents its limit?

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Yes, that would be correct. In other words: if $(a_n)_{n\in\mathbb N}$ is a Cauchy sequence which is convergent in $\mathbb R$ (note that this is equivalent to the assertion that it is a Cauchy sequence), then its limit $l$, seen as Dedekind cut is precisely the set$$\{x\in\mathbb Q\,|\,(\exists n\in\mathbb N)(\forall m\in\mathbb N):m\geqslant n\implies x<a_m\}$$(I used $\leqslant$ here instead of $<$ just for a matter of taste).

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