Question 99
An infinite sequence $(a_n)$, $n \in \mathbb{N}$, of positive numbers is called *lacunar* if there exists a number $q \gt 1$ such that $a_{n+1}/a_n \ge q$ for all $n \in \mathbb{N}$. Also, the sequence is called *rare* if there exists a positive integer $k$ such that the interval $(x, 2x)$ contains at most $k$ terms of this sequence for any positive $x$.
a) Is it true that any lacunar sequence is rare?
b) Is it true that any increasing rare sequence is lacunar?
a) Is it true that any lacunar sequence is rare?
b) Is it true that any increasing rare sequence is lacunar?