Problem 10.5 (first part). Let $\mathbb{K}$ be a real number field. Then, for any $\varepsilon > 0$, there exists a lacunary sequence $(t_n){n \ge 1}$ of positive numbers in $\mathbb{K}$ such that $$\limsup{n \to \infty} {\xi t_n} \ge 1 - \varepsilon,$$ for any real number $\xi$ not in $\mathbb{K}$.

Problem 10.5 ("moreover" clause). With the same hypotheses as problem_10_5, the sequence $(t_n)$ can be chosen so that, for any real $\xi$ not in $\mathbb{K}$, each subinterval of $[0, 1]$ of length $\varepsilon$ contains a limit point of the sequence $({\xi t_n})_{n \ge 1}$. This is strictly stronger than problem_10_5: the limsup bound is the special case at the subinterval $[1 - \varepsilon, 1]$.

The "moreover" form of Problem 10.5 implies the first part: applying the cluster-point density to the subinterval $[1 - \varepsilon, 1]$ yields the required lower bound on the limsup.