Question 52
Prove that there exists a real number $c \lt 3/4$ such that for each sequence $\{x_i\}_{i=1}^\infty$ satisfying $0 \le x_i \le 1$ for all $i$, there are infinitely many pairs $(m, n)$ with $m \gt n$ such that
$$
|x_m - x_n| \le \frac{c}{m}.
$$
$$
|x_m - x_n| \le \frac{c}{m}.
$$