$A$ is a discrete valuation ring, $m$ its maximal ideal, $\hat A, \hat m$ their completion respectively.
In J.S. Milne's Algebraic Number Theory, p. 116:
Why the openness implies the surjective part?
What i think so far, given $x \in \hat A$, since $\hat A$ is an completion, there exists $a\in A$, such that $x-a=\pi^tu$ with $t$ big and $u \in \hat A \setminus \hat m^n$ which is closed, but I don't know how to proceed.