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Lemma 10 ([Continuity within a fixed interval]). Suppose I⊆X_k⊆Y for k≥1 , where I,Y∈ C(T) , and membership of every element in X_k eventually agrees with its membership in a set X . Then X∈ C(T) and, for every fixed n≥1 , f_n^T(X_k)⟶f_n^T(X).