Modules over local ring and completion I'm stuck again at a commutative algebra question. Would love some help with this completion business...
We have a local ring $R$ and $M$ is a $R$-module with unique assassin/associated prime the maximal ideal $m$ of $R$. 
i) prove that $M$ is also naturally a module over the $m$-adic completion $\hat{R}$, and $M$ has the same $R$ and $\hat{R}$-submodules.
ii) if $M$ and $N$ are two modules as above, show that $Hom_{R}(M,N) = Hom_{\hat{R}}(M,N)$. 
Best wishes and a happy new year!

Here's a claim from which all these statements would follow: For any Cauchy sequence $r_i \in R$, and $x \in M$, the sequence $r_i x$ is eventually constant.  (The reason for believing this claim is that, for example, for the first part of i) we have to take a "limit" of $r_i x$ while remaining in $M$, and it's hard to see how else one could do this with no topology on $M$.)
Can we prove this claim?  First, let's try this: Since $m$ is an associated prime of $M$, we know that for some $x \in M$, the annihilator of $x$ is $m$.  For such an $x$, we can see that $r_i x$ is eventually constant because the Cauchyness of the $r_i$ implies that for some large $N$, we have $r_i - r_j \in m$ for all $i,j \ge N$, hence $r_i x = r_j x$ for all $i,j \ge N$.  Unfortunately, this argument doesn't cover all $x \in M$.
However, for a general $x$, if we can show that ann($x$) contains a power of $m$, then we could still apply the same argument as above.  Somehow this should follow from the fact that $m$ is the only associated prime ideal of $M$ (and we may have to assume that $R$ is Noetherian).  Edit: See YACP's answer for a proof that ann($x$) contains a power of $m$.
