Variant of Nakayama's lemma 
I am trying to prove that if $M$ is an $R$-module, with $R$ complete w.r.t. an ideal $\mathfrak{m}$, and $M$ is separated ($\cap_k \mathfrak{m}^k M=0$) and the images of $m_1,\dots,m_n$ generate $M/\mathfrak{m} M$, then $m_1,\dots,m_n$ generate $M$. 

This appears as Exercise 7.2 in Eisenbud's Commutative Algebra text. 
I am pretty stuck and would appreciate some hints. 

Hint. Set $N=\langle m_1,\dots,m_n\rangle$. We have $M=\mathfrak mM+N=\mathfrak m(\mathfrak mM+N)+N=\mathfrak m^2M+N$, and so on.
