# Cardinality of a union of ordinal-indexed nested sets with bounded cardinality?

Let $$(X_\alpha)$$ be a nested ordinal-indexed sequence of sets, i.e. if $$\alpha\leq\beta$$ then $$X_\alpha\subseteq X_\beta$$ for all ordinals $$\alpha$$ and $$\beta$$, and suppose that there is a cardinal $$\kappa$$ such that $$|X_\alpha|\leq\kappa$$ for all ordinals $$\alpha$$. My question is, what can we say about the cardinality of $$\cup_\alpha X_\alpha$$?

First of all $$\cup_\alpha X_\alpha$$ is a set as opposed to a proper class? Second of all, can we put some kind of bound on the cardinality of $$\cup_\alpha X_\alpha$$?

This came up in the course of writing up a proof of the downward Lowenheim-Skolem theorem, by the way.

Since the chain is well-ordered, it is necessarily stabilized by some $$\gamma<\kappa^+$$, otherwise choosing some $$x_\alpha\in X_{\alpha+1}\setminus X_\alpha$$ for $$\alpha<\kappa^+$$, would define an injection from $$\kappa^+$$ into $$X_{\kappa^+}$$.
So indeed the union is a set, and its cardinality is at most $$\kappa$$. And easily, for any cardinal $$\lambda\leq\kappa$$, the union could have cardinality $$\lambda$$ (which means that it may have stabilized before $$\kappa$$).