A sequence $\{a_i\}$ is given such that $a_1 = \frac13$ and for all positive integers $n$ $$a_{n+1} =\frac{a^2_n}{a^2_n - a_n + 1}.$$Prove that $$\frac12 - \frac{1}{3^{2^{n-1}}} < a_1 + a_2 +... + a_n <\frac12 - \frac{1}{3^{2^n}} ,$$for all positive integers $n$.