Since the sequence is bounded below by Xi > X. Hence the limit
exists and is hermitian, and we have X+ ³ X. Passing the limit n®¥ in (0.9), we get Q(X+)=0. So X+ is a solution of
(0.7).
Note that X+ -X ³ 0 and
(X+-X)A++A+*(X+-X)=-Q(X )+(X+-X)*BB*(X+-X)>0
Hence, X+ -X>0 and A+=A+BB*X + is antistable.