The Integers Lying in a Bounded Real Interval Form a Finite Set

In the setting of Routine Facts Used Without Citation: Arithmetic, Maps, Predicativity and Indices, let $\mathbb{R}$, its order $\le$ and its operations be as in The Real Numbers, with the Natural Numbers, Integers and Rationals Identified with Subsets of the Reals, and Completeness (ii), let $\mathbb{Z}$ and its order be as in The Integers and the Rational Numbers, with the Natural Numbers and the Integers Identified with Subsets of the Rationals (ii), let $j:\mathbb{Z}\to\mathbb{Q}$ be the embedding of The Integers and the Rational Numbers, with the Natural Numbers and the Integers Identified with Subsets of the Rationals (v) and $\kappa:\mathbb{Q}\to\mathbb{R}$ the canonical embedding of The Real Numbers, with the Natural Numbers, Integers and Rationals Identified with Subsets of the Reals, and Completeness (iv), and let finite sets be as in Finite Sets (i). Let $a,b\in\mathbb{R}$.

(i) There is a finite subset $E$ of $\mathbb{R}$ such that, for every $u\in\mathbb{R}$, $u\in E$ if and only if $u=\kappa(j(n))$ for some $n\in\mathbb{Z}$ with $a\le\kappa(j(n))\le b$.