The $p$-adic closure of the global units lies in $\bigcap_n \iota(E)U^{p^n}$

Let $K$ be a number field, $p$ a prime, $U=\prod_{\mathfrak p\mid p}\mathcal O_{\mathfrak p}^\times$ the semilocal units at $p$ and $\iota:E=\mathcal O_K^\times\to U$ the diagonal embedding. Then the topological closure of $\iota(E)$ in $U$ is contained in

This is the inclusion $\overline{\iota(E)}\subseteq\bigcap_n\iota(E)U^{p^n}$ of the identity $\bar E=\overline{\iota(E)}=\bigcap_n\iota(E)U^{p^n}$ in §1.1 of the source; it follows because the right-hand side is a closed subgroup containing $\iota(E)$. (The reverse inclusion can fail by a finite group of roots of unity of order prime to $p$ — e.g. for $K=\mathbb Q$, $p=7$ the intersection is $\mu_6$ while $\overline{\iota(E)}=\{\pm1\}$ — which does not affect $\mathbb Z_p$-ranks.)