recall that Pick any ( y ′ $_{0}$, y ′ $_{1}$) with y ′ 1 > y ′ 0 in the optimal support of the joint distribution. For any h s.t. 0 < h < y ′ $_{1}$− y ′ $_{0}$, The inequality in the second line of (A.13) is satisfied because y ′ 1 > ( y ′ 0 + h ) , and the inequality in the third line of (A.13) holds since 1 { y$_{1}$ − y$_{0}$ > δ } is nonincreasing in y$_{0}$. By Lemma A.2, ϕ is nonincreasing on$_{R}$. Now, B$_{k}$ = { y ∈$_{R}$ ; ϕ > s + k } = ( −∞ , b$_{k}$ ) for each integer k , some s ∈ (0 , 1] and b$_{k}$ ∈ [ −∞ , ∞ ] , in which B$_{k}$ = φ for b$_{k}$ = −∞ . By Theorem 1, for each integer k, b$_{k}$$_{+1}$ ≤ b$_{k}$ and for δ > 0 , If b$_{k}$ = −∞ , then b$_{k}$$_{+1}$ = −∞ and so B D k = φ. For b$_{k}$ > −∞ , B D k depends on the value of b$_{k}$$_{+1}$ as follows: ϕ ( y$_{0}$ ) = inf y$_{1}$ ≥ $_{y$_{0}$}${{ y$_{1}$ − y$_{0}$ > δ } − ψ ( y$_{1}$ ) } . ϕ ( y ′ 0 + h ) = inf y$_{1}$>y ′ $_{0}$+ h { 1 { y$_{1}$ − ( y ′ 0 + h ) > δ } − ψ ( y$_{1}$ ) } (A.13) ≤ 1 { y ′ 1 − ( y ′ 0 + h ) > δ } − ψ ( y ′ $_{1}$) ≤ 1 { y ′ 1 − y ′ 0 > δ } − ψ ( y ′ $_{1}$) = ϕ ( y ′ $_{0}$) , B D k = { y$_{1}$ ∈$_{R}$ ; ∃ y$_{0}$ < b$_{k}$ s.t. 0 ≤ y$_{1}$ − y$_{0}$ < δ } ∪ { y$_{1}$ ∈$_{R}$ ; ∃ y$_{0}$ < b$_{k}$$_{+1}$ s.t. y$_{1}$ − y$_{0}$ ≥ δ } . B D k =      $_{R}$, if b$_{k}$$_{+1}$ > −∞ , ( −∞ , b$_{k}$ + δ ), if b$_{k}$$_{+1}$ = −∞ . 67 Figure A.4: B D k for B$_{k}$ = ( −∞ , b$_{k}$ ) and B$_{k}$$_{+1}$ = ( −∞ , b$_{k}$$_{+1}$ )