Completion and completeness

Question

Let $\widehat{M}$ be the completion of a module $M$ via a filtration $\mathcal{F}=\{M_i:i\in\mathcal{I}\}$ where $\mathcal{I}$ is a directed set. I understand the completion as: $$\varprojlim_{i\in\mathcal{I}} M/M_i$$ as the latter one I think as coherent sequences: $(x_i)_{i\in\mathcal{I}}$ with $\theta_{n+1}(x_{n+1})=x_n$. I was trying to explore why is $\widehat{M}$ topologically complete. A lot of people at this point will be satisfied with saying if $M\simeq \widehat{M}$ then it is complete, but this is just a definition. We can show that $\widehat{M}$ is topogically complete by checking at Cauchy sequences in $\widehat{M}$.

My idea: If $((a_{ij}))_{i,j\in\mathcal{I}}$ is a Cauchy sequence of coherent sequences, then I showed the $a_{ij}$ form a direct system for a fixed i, hence one can define: $$a_i=\varinjlim_{i\in\mathcal{I}}a_{ij}$$ and claim that $$((a_{ij}))_{i,j\in\mathcal{I}}\rightarrow (a_i)_{i\in\mathcal{I}}$$ I'm given the coherent sequences the subspace topology of the product topology om $M/M_i$, i.e., basic open sets look like $\prod U_i\cap \widehat{M}$, where almost all $U_i$ are $M/M_i$ and the others look like $M_\ell/M_i$, where $\ellN$ then $(a_{ij})-(a_i)\in \prod U_i \cap \widehat{M}$. Can anyone show me how to use the Cauchyness to prove that this is in fact the limit? Is this something worth thinking of? Haven't seen it in any book so far.