# Dedekind cut corresponding to the limit of a Cauchy sequence

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?

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$$\{x\in\mathbb Q\,|\,(\exists n\in\mathbb N)(\forall m\in\mathbb N):m\geqslant n\implies x(I used $$\leqslant$$ here instead of $$<$$ just for a matter of taste).