Canonical isomorphism between Cauchy sequence completion and inverse limit I'm studying chapter 10 of Atiyah Macdonald. The book introduces two ways to construct the completion of an abelian topological group: Equivalence classes of Cauchy sequences and inverse limit. I can see how these two are isomorphic as groups. However, the book doesn't explain how they're topologically equivalent (homeomorphic) and I'm unable to fill in the details. I'm looking for a reference that does so.
I've checked a few references. Most details are skipped. (Is it really that easy? I can't see it.) There are also many variations: Some take equivalence classes of Cauchy sequences modulo null sequences. The topology defined on the completion can also be expressed differently.
Here is how the book defines a Cauchy sequence in an abelian topological group:

$ (x_n) $ is a Cauchy sequence if for each neighbourhood $ U $ of $ 0 $, there is $ N $ such that $ n, m > N $ means $ x_n - x_m \in U$.

Given the definition above, I prefer a reference that doesn't use nets or metrics.
Given a chain of subgroups $G = G_0 \supset G_1 \supset G_2 \supset ...$, the book gives $ G $ the topology in which $\langle G_{n} \rangle $ is a neighbourhood basis of $ 0 $. The inverse limit considered is then $ \varprojlim_n G / G_n $.
Thanks

Outline of the proof.
Let $C$ be the group of Cauchy sequences (without the equivalence classes.)
If $\mathbf{a}=\{a_i\}$ is Cauchy, define $N_k(\mathbf {a})$ the be the least value so that for all $i,j\geq N_k(\mathbf a),\ a_i-a_j\in G_k$. This can be written as $a_i+G_k=a_j+G_k$ in $G/G_k$. Then define $\phi_k:C\to G/G_k$ by $\{a_i\}\to a_{N_k(\mathbf{a})}+G_k$.  Show that this is well-defined, and map satisfies the inverse limit criterion. If $p_k:G/G_{k+1}\to G/G_k$ we need to show:
$$p_k\circ\phi_{k+1}=\phi_k$$
So the universal property shows that there is a homomorphism $\phi: C\to\varprojlim_n G / G_n$.
The next step is the to prove that the kernel is exactly the Cauchy sequences that are considered zero in your definition. I think that's easy.
So this shows an isomorphism of groups between these two constructs, but then you need to prove that they have the same open sets.
