In the present paper we shall prove that the limit of a monotonously increasing sequence of Cousin-I domains over a Stein manifold is a Cousin-I domain. Concerning the Cousin-II problem, however, we can prove that the limit of the monotonously increasing sequence of Cousin-II domains over a Stein manifold is a Cousin-II domain, only in case that it is simply connected. The proof is based on the theory of domains of holomorphy due to and the approximation theory due to Behnke [1].
As Fuks [3] stated, every domain of holomorphy or meromorphy over Cn is analytically convex in the sense of Hartogs. Oka [6] proved that every domain over Cn analytically convex in the sense of Hartogs is a domain of holomorphy. Therefore a domain of meromorphy over Cn coincides with a domain of holomorphy over Cn.
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