Erdős Problem #280 — Uncovered Integers in Congruence Systems

Let $n,a:\mathbb{N}\to\mathbb{N}$ satisfy that $n$ is strictly increasing and $a(i)<n(i)$ whenever $i\geq1$. For $k\in\mathbb{N}$, let $U_k$ be the number of natural numbers $m<n(k)$ that are not covered by any of the first $k$ congruence classes, meaning that there is no $i\in\{1,\ldots,k\}$ with

$$m\equiv a(i)\pmod{n(i)}.$$

If there exists $\varepsilon>0$ such that, for every $k\geq1$,

$$n(k)>(1+\varepsilon)k\log k,$$

must

$$\frac{U_k}{k}$$

fail to converge to $0$ as $k\to\infty$?