Since you edited your question, I will post a different answer. You know that $\mathfrak m^i$ is a finitely generated ideal because $A$ is Noetherian. Let $m_1, \dots, m_n$ be a finite set of generators of $\mathfrak m^i$ as an $A$-module. Try to show that the images of these generators of $m^i$ under the canonical projection map $\mathfrak m^i \rightarrow \mathfrak m^i/\mathfrak m^{i+1}$ generate $\mathfrak m^i/\mathfrak m^{i+1}$ as an $A/\mathfrak m$-module (after all what is the $A/\mathfrak m$-module action on $\mathfrak m^i/\mathfrak m^{i+1}$?). Now $A/\mathfrak m$ is a field, since $\mathfrak m$ is a maximal ideal. What is a module over a field?