Surjectivity on adic completions ($M\twoheadrightarrow N⟹\hat{M}\twoheadrightarrow \hat{N}$)

For any ideal $I$ and surjective module map $f:M\to N$, compatible representatives $n_r$ modulo $I^{r}N$ can be lifted recursively. Having chosen $f(m_r)=n_r$, lift $n_{r+1}-n_r\in I^{r}N$ to ${\delta }_r\in I^{r}M$ and put $m_{r+1}=m_r+{\delta }_r$. These define the required element of the adic completion of a module. No finite-generation hypothesis on the modules is required.