A fundamental classical theorem states that every compact orbit of a dynamical system is periodic. We firrst discuss the constructively inadmissible aspects of the standard classical proof, to reveal some of the non-trivial problems associated with the search for a (Bishop-style) constructive proof. We then strip away the apparatus of dynamical systems, to reveal the problem as one about continuous epimorphisms from R to metric abelian groups. There are then two classically equivalent but constructively independent theorems: Theorem 1 Let h be a continuous homomorphism of R onto a compact abelian group G. Then there exists t<> 0 such that h(t) = 0. Theorem 2 Let h be a continuous one-one homomorphism of R onto a complete metric abelian group G, such that the set S_1 ={h(t) : |t| > 1} is located in G. Then G is noncompact, in the sense that for each compact K \subset G, the metric complement G - K :={x\in G : ho(x; K) > 0} is inhabited. The proof in each case uses Baire's category theorem. For Theorem 2 we need to develop a number of classically vacuous, constructively interesting (and even amusing) preliminary results about injective mappings from R onto a complete metric space. This is joint work with Matt Hendtlass.