Continuous finite-dimensional representations of profinite groups factor through finite quotients

If $n=0$, $U(0)$ is the trivial group, $\ker \rho =K$ and the representation factors through the trivial finite quotient; also $\ker {\pi }_i\subseteq K$ for any index $i$. Hence assume $n\ge 1$. First $\rho$ is continuous in operator norm: for an orthonormal basis $e_1,\dots ,e_n$ of ${\mathbb{C}}^n$ and $x={\sum }_i{x}_i{e}_i$ with $\mathrm{\parallel }x\mathrm{\parallel }\le 1$, $\mathrm{\parallel }(\rho (k)-\rho (k_0))x\mathrm{\parallel }\le {\sum }_i\mathrm{\mid }{x}_i\mathrm{\mid }\mathrm{\parallel }(\rho (k)-\rho (k_0))e_i\mathrm{\parallel }\le ({\sum }_i\mathrm{\parallel }(\rho (k)-\rho (k_0))e_i{\mathrm{\parallel }}^2{)}^{1\mathrm{/}2}$ by Cauchy–Schwarz, and the right side tends to $0$ as $k\to k_0$ because the finitely many orbit maps are continuous [F3]; by [F2] the group $U(n)$ is a finite-dimensional real Lie group, so its no-small-subgroups lemma provides an open identity neighbourhood $V\subseteq U(n)$ containing no subgroup other than $\{e\}$, and $W:={\rho }^{-1}(V)$ is an open neighbourhood of the identity of $K$.

By [F1] the subgroups $\ker {\pi }_i$ form an open normal neighbourhood basis at the identity, so $\ker {\pi }_i\subseteq W$ for some $i$; then $\rho (\ker {\pi }_i)$ is a subgroup of $U(n)$ by [F4] and lies in $V$, hence $\rho (\ker {\pi }_i)=\{e\}$ by the choice of $V$, that is, $\ker {\pi }_i\subseteq \ker \rho$. Since $\ker {\pi }_i$ is open, its cosets are open, and $\ker \rho$ is a union of cosets of $\ker {\pi }_i$, so $\ker \rho$ is open as well.

The inclusion $\ker {\pi }_i\subseteq \ker \rho$ implies that $\rho$ factors as $\rho =\bar{\rho }\circ q$, where $q:K\to F:=K\mathrm{/}\ker {\pi }_i$ is the quotient homomorphism and $\bar{\rho }(q(k)):=\rho (k)$; here $q$ is a surjective continuous homomorphism and $F$ is finite because the first isomorphism theorem gives $F\cong {\pi }_i(K)\subseteq G_i$ with $G_i$ finite [F4]. Every matrix coefficient $c_{v,w}^{\rho }$ is constant on each coset of $\ker {\pi }_i$, since $\rho$ is, hence is locally constant and factors through the finite group $F$; this proves the lemma. The Axiom of Choice is consumed through the countable choice assumed by the Lie-group example [F2] and the profinite presentation; the factorisation argument itself is choice-free apart from those inputs.