1. As x ? X 1 , the argument that we have used to establish (59) in Step 2 yields u 1 (x ) > u 2 (x ) and thus u 3 (x ) = u 1 (x ), establishing (56) for the case under consideration;Consider (x , Y ) ? Supp;By construction of ? 3 we then have (x , y ) ? supp(? 1 ), implying u 1 (x ) = ?(x , y , v 1 (y ))
2. As shown above u 3 (x n ) = ?(x n , y n , v 3 (y n )) holds for all n in this sequence. As ?, v 3 and u 3 are all continuous, the convergence of (x n , y n ) ? n=1 to (x , y ) implies u 3 (x ) = ?(x , y , v 3 (y )), which is the desired result. It remains to show that the set of pairwise stable outcomes for the matching model;� Y 1 ) and supp(? 3 ) ? (X � Y 2 ), there then exists a sequence (x n , y n ) ? n=1 in this union which converges to (x , y )