a_{n+2} = a_{n+1}^2 + \displaystyle\frac{1}{6}a_n +\displaystyle\frac{1}{9} \leq \displaystyle\frac{1}{36} + \displaystyle\frac{1}{36} + \displaystyle\frac{1}{9} = \displaystyle\frac{1}{6}