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By part (i), finite-dimensional continuous unitary representations separate points of the compact group. More explicitly, if every representation sent some $g\ne e$ to the identity, all their matrix coefficients would take the same values at $g$ and $e$. Uniform density would force every continuous scalar function to do so, contradicting separation of points on a compact Hausdorff space. HenceEach representation kernel is a closed normal subgroup. We claim that the intersection can be reduced to finitely many kernels. Begin with $K_0=G$. If $K_j\ne \{e\}$, choose $g_j\in K_j\setminus \{e\}$ and a representation ${\theta }_{j+1}$ with ${\theta }_{j+1}(g_j)\ne I$. ThenAn indefinite continuation would violate the descending chain condition for closed subgroups. Thus after finitely many steps, $K_N=\{e\}$.

The direct sum $\rho ={\theta }_1\oplus \cdots \oplus {\theta }_N$ is consequently a finite-dimensional faithful representation, with continuous injective homomorphismCompactness of $G$ makes this a homeomorphism onto a compact, hence closed, image. The closed-subgroup theorem makes that image an embedded Lie subgroup of the finite-dimensional unitary group. Transporting its smooth structure to $G$ provesThis is the mechanism behind the assertion that compact groups with a descending chain condition are Lie groups: the chain condition turns an arbitrarily large separating family into one finite-dimensional faithful representation.