Figure 8: A D k for A$_{k}$ = ( a$_{k}$, ∞ ) and A$_{k}$$_{+1}$ = ( a$_{k}$$_{+1}$, ∞ ) . For any ( y ′ $_{0}$, y ′ $_{1}$) and ( y ′′ $_{0}$, y ′′ $_{1}$) with y ′′ 0 > y ′ 0 in the support of the optimal joint distribution, The first inequality in the second line follows from y ′′ 1 ≥ y ′′ 0 > y ′ 0 The second inequality in the third line is satisfied because 1 { y$_{1}$ − y$_{0}$ < δ } is nondecreasing in y$_{0}$. Consequently, ϕ is nondecreasing and thus A$_{k}$ = ( a$_{k}$, ∞ ) for a$_{k}$ ∈ [ −∞ , ∞ ] at the optimum. A D k is obtained from A$_{k}$ as follows: for δ > 0 and A$_{k}$ = ( a$_{k}$, ∞ ) and A$_{k}$$_{+1}$ = ( a$_{k}$$_{+1}$, ∞ ) , At the optimum, { a$_{k}$ } ∞ k = −∞ should satisfy a$_{k}$$_{+1}$ ≤ a$_{k}$ + δ for each integer k . The rigorous proof is provided in Appendix A. I demonstrate this graphically here. As shown in Figure 7(c), my improved lower bound represents the sum of Fr´ echet lower bounds on the probability of a sequence of disjoint triangles. Suppose 22 A D k = { y$_{1}$ ∈$_{R}$|∃ y$_{0}$ > a$_{k}$ s.t. δ ≤ y$_{1}$ − y$_{0}$ } ∪ { y$_{1}$ ∈$_{R}$|∃ y$_{0}$ > a$_{k}$$_{+1}$ s.t. 0 ≤ y$_{1}$ − y$_{0}$ < δ } = ( a$_{k}$ + δ, ∞ ) ∪ ( a$_{k}$$_{+1}$, ∞ ) = (min { a$_{k}$ + δ, a$_{k}$$_{+1}$ } , ∞ ) . ϕ ( y ′ $_{0}$) = 1 { y ′ 1 − y ′ 0 < δ } − ψ ( y ′ $_{1}$) ≤ 1 { y ′′ 1 − y ′ 0 < δ } − ψ ( y ′′ $_{1}$) ≤ 1 { y ′′ 1 − y ′′ 0 < δ } − ψ ( y ′′ $_{1}$) = ϕ ( y ′′ $_{0}$) .