Externalities, Nonconvexities, and Fixed Points

10 Appendix 2: The Proof of Lemma 8(1) Let {E n } n ⊂ S be a sequence of Ky Fan sets such that h w ∗ ×w ∗(En ,E 0 ) → 0. Wemust show that E 0 ∈ S. Becauseh w ∗(D(E n ), D(E 0 )) → 0and because each D(E n )=R(E n ) is convex, we have that D(E 0 ) ∈ P w ∗ fc(X) andD(E 0 )=R(E 0 ). Thus, E 0 satisfies (Z1). Also, note that E 0 satisfies (Z2). Theproof will be complete if we can show that for all x ∈ R(E),{y ∈ D(E) :(y, x) /∈ E}is convex and possibly empty (i.e., that E 0 satisfies (Z3)). Suppose not. Then forsome y 1 , y 2 ,andx 0 in R(E 0 ),wehavey i ∈ © y ∈ D(E 0 ):(y,x 0 ) /∈ E 0ª , i =1, 2, (*)but for some λ 0 ∈ (0, 1), y 0 = λ 0 y 1 +(1− λ 0 )y 2 ∈ D(E 0 ) but(y 0 ,x 0 ) ∈ E 0 . (**)Because D(·) is continuous (and in particular, lower semicontinuous) there exist sequences,{y 1n } n and {y 2n } n , such that y in ∈ D(E n ) for all n, i =1, 2, andy 1n → y1 ,w ∗y 2n → y2 .w ∗Therefore,y 0n = λ 0 y 1n +(1− λ 0 )y 2n →w ∗ y0 ∈ D(E n )We have (y 1 ,x 0 ) /∈ E 0 , (y 2 ,x 0 ) /∈ E 0 , but (y 0 ,x 0 ) ∈ E 0 .Becausewe have for all n sufficiently large,(y in ,x 0 ) →w ∗ ×w ∗ (yi ,x 0 ) /∈ E 0 , i =1, 2,andh w ∗ ×w ∗(En ,E 0 ) → 0,(y in ,x 0 ) /∈ E n , i =1, 2.Because E n ∈ S for all n, wehaveforalln sufficiently large,(y 0n ,x 0 )=(λ 0 y 1n +(1− λ 0 )y 2n| {z }y 0n ,x 0 ) /∈ E n .Thus,(y 0n ,x 0 ) ∈ [D(E n ) × X] ∩ [(X × X)\E n ] for all n.50