A Product Threshold

In a sequence $\{a_n\}$, $a_1 = \dfrac{2}{3}$, $a_2 = \dfrac{8}{9}$, and for $n \geqslant 2$,

1. Prove that $\{a_{n+1} - a_n\}$ is geometric. 2. Find a formula for $a_n$. 3. If $\lambda a_1a_2\cdots a_n \geqslant 1$ for every $n \in \mathbb{N}^*$, where $\lambda \in \mathbb{N}^*$, find the minimum value of $\lambda$.