The infinite sequence $a_0, a_1, a_2, a_3,... $ is defined by $a_0 = 2$ and $$a_n =\frac{2a_{n-1} + 1}{a_{n-1} + 2}$$, $n = 1, 2, 3, ...$ Prove that $1 < a_n < 1 + \frac{1}{3^n}$ for all $n = 1, 2, 3, . .$