Let $A$ be a commutative complete ring with unit for the $I$-adic topology, where $I$ is the ideal of $A$. Let $(M_n)_{n\geq 0}$ be $A$-modules such that $I^{n+1}M_n=0$ and that there exist a surjective homomorphism $\pi_n:M_{n+1}\to M_n$ with $\ker (\pi_n)=I^{n+1}M_{n+1}$. Let $M=\varprojlim_nM_n$ and denote the surjective canonical homomorphism by $u_n:M\to M_n$. Let us suppose that $M_0$ is generated over $A$ by finite number of elements $e_{0,1},\ldots,e_{0,m}$. Let $e_1,\ldots,e_m\in M$ be such that $u_0(e_i)=e_{0,i}$, and define $\phi_n:A^m\to M_n$ by $(a_1,\ldots,a_m)\mapsto \sum_i a_iu_n(e_i)$. How can I show that $\phi_n$ is surjective and that $M$ is generated by the $e_i$?
Qing Liu: Algebraic Geometry and Arithmetic Curves ex 1.3.11 b
