Question 7
The sequence $(a_n)_{n \ge 1}$ has the properties that $a_n \gt 1$ and $a_{n+1}^2 \ge a_n a_{n+2}$, for $n \ge 1$. Show that the sequence $(x_n)_{n \ge 1}$, defined by $x_n = \log_{a_n} a_{n+1}$, for $n \ge 1$, has a finite limit. Find it.