Union of Saturated Models in Superstable Abstract Elementary Classes

In this paper we prove: Theorem 1. Let $\mathcal{K}$ be an abstract elementary class which satisfies the joint embedding and amalgamation properties. Suppose $λ>μ\geq LS(\mathcal{K})$ and $θ$ is a limit ordinal $<λ^+$. If $\mathcal{K}$ is $μ$ superstable and $μ^+$-superstable and satisfies $μ^+$-symmetry, then for any increasing sequence $\langle M_i\mid i<θ\rangle$ of $μ^+$-saturated models of cardinality $λ$, the model $\bigcup_{i<θ}M_i$ is $μ^+$-saturated.

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