Every Cauchy sequence is definable in a d-minimal expansion of the $\mathbb R$-vector space over $\mathbb R$

Every Cauchy sequence is definable in a d-minimal expansion of the $\mathbb R$-vector space over $\mathbb R$. In this paper, we prove this assertion and the following more general assertion: Let $\mathcal R$ be either the ordered $\mathbb R$-vector space structure over $\mathbb R$ or the ordered group of reals. A first-order expansion of $\mathcal R$ by a countable subset $D$ of $\mathbb R$ and a compact subset $E$ of $\mathbb R$ of finite Cantor-Bendixson rank is d-minimal if $(\mathcal R,D)$ is locally o-minimal.