Let (X, d) be a compact metric space and let A ⊆ C(X; C) = {f : X → C; f is continuous} be an algebra that separates points and vanishes at no point in X. Assume additionally that A is self-adjoint, that is, for every f ∈ A, its complex conjugate f¯ is also in A. Show that A is dense in C(X; C).

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