Let $\mathbb{N}^*$ be the set of positive integers. Define $a_1=2$, and for $n=1, 2, \ldots,$\[ a_{n+1}=\min\{\lambda|\frac{1}{a_1}+\frac{1}{a_2}+\cdots\frac{1}{a_n}+\frac{1}{\lambda}<1,\lambda\in \mathbb{N}^*\}\] Prove that $a_{n+1}=a_n^2-a_n+1$ for $n=1,2,\ldots$.