Question 191
Show that there exist a sequence $(a_n)_{n \ge 0}$, with $a_n \in \{-1, +1\}$ for all $n \ge 0$, such that
$$
\lim_{n \to \infty} (\sqrt{n+a_1} + \sqrt{n+a_2} + \dots + \sqrt{n+a_n} - n\sqrt{n+a_0}) = \frac{1}{2}.
$$
$$
\lim_{n \to \infty} (\sqrt{n+a_1} + \sqrt{n+a_2} + \dots + \sqrt{n+a_n} - n\sqrt{n+a_0}) = \frac{1}{2}.
$$