I am aware of the question (Topology on $\mathbf{Z}_p$)
One can use two definitions of the p-adic integers, the one that sees them as infinite formal sums $\sum_{i=0}^\infty a_i p^i$ and the one that sees them as inverse limit $\varprojlim \mathbb Z/p^i\mathbb Z$. One can show that those two are isomorphic as rings.
Now the infinite formal sums can be endowed with the p-adic norm, which gives rise to a topological structure induced by the norm; and the inverse limit inherits the product topology (we give each group the discrete topology).
I am trying to show that those two topologies are equal.
So given a
The p-adic norm induces a metric in which a neighbourhood base of $0$ is given by the open balls $\{B(0,\frac{1}{p^n})\}$. Infinite formal sums in $B(0,\frac{1}{p^n})$ are divisible by $p^n$, and henceforth correspond to sequences in the inverse limit belonging to the subset $\left(\underbrace{\{0\}\times \{0\}}_{n\text{ times}}\times \prod_{k\geq n+1} \mathbb Z/p^k\mathbb Z\right)\cap \varprojlim \mathbb Z/p^i\mathbb Z$, obviously an open set.
Is this correct?
Conversely, an open basis set in $(\mathbb Z_p)_{\text{proj}}$ is of the form $$\left(\prod_{j\in K\subset \mathbb N\text{ finite}} U_j \times \prod_{j\in \mathbb N\setminus K} \mathbb Z/p^j\mathbb Z\right)\cap \varprojlim \mathbb Z/p^\ell \mathbb Z$$ where $U_j\subset \mathbb Z/p^j\mathbb Z$ is open .
How do I show that this corresponds to an open set in the p-adic metric?