KöMaL 2006 A395

Let 1<$a$<2 be a real number. ($a$) Show that there exists a unique sequence $x_1$,$x_2$,... of positive integers satisfying $x_{i+1} \ge x_i ^2$ for all indices $i$ and $\left(1+\frac1{x_1}\right)\left(1+\frac1{x_2}\right)\dots = a.$ ($b$) Prove that inequality $x_{i+1}$>$x_i ^2$ holds for infinitely many indices if and only if $a$ is irrational.