Proposition

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 , …

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

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Lemma 9 ([Probe properties]). For X,Y∈ C(T) the following statements hold.…Lemma 8 ([Simultaneous reservations]). Let S be a family of fewer than c…Proposition
We have I⊆X⊆Y and Y I∈ J_0 , so X∈ C(T) . On every fixed finite union of blocks, X_k eventually agrees with X . On the remaining blocks, |d_m^T(X_k)-d_m^T(X)| ≤(|(Y I)∩W_m|)/(|W_m|), uniformly in k , and the right-hand side tends to zero. Thus the sequences of block densities converge uniformly in m . Supremum and infimum over a fixed tail change by at most the uniform difference, while the fixed prefix eventually stabilizes. Formula gives the result.
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