Continuity of the Galois action on semilocal units

Let $K$ be a number field, $p$ a rational prime, $\mathcal P$ the set of primes of $\mathcal O_K$ above $p$, and $U=\prod_{v\in\mathcal P}\mathcal O_v^\times$ the group of semilocal units at $p$ with its product (profinite) topology. For $\sigma\in\operatorname{Aut}(K/\mathbb Q)$ let $\sigma_U:U\to U$ be the Galois action on semilocal units, $(\sigma_U x)_w=\sigma(x_{\sigma^{-1}w})$, where $\sigma:K_{\sigma^{-1}w}\xrightarrow{\sim}K_w$ is the continuous extension of $\sigma$ to completions (definition Def_LeopoldtGaloisAction).

Theorem. For every $\sigma\in\operatorname{Aut}(K/\mathbb Q)$ the automorphism

is continuous.

Together with the group-action property (so that $\sigma_U^{-1}=(\sigma^{-1})_U$ is continuous too) this makes $\sigma_U$ a topological group automorphism, so the action preserves closed subgroups, $\mathbb Z_p$-ranks and continuous injections $\mathbb Z_p^n\hookrightarrow U$.