Generalization of Fatou's Lemma Question in the proof of Royden [closed]

In this proof given by Royden in Real Analysis textbook, I have a hard time understanding why in case 1, the set $X_{\infty} \subseteq \cup_{n\in N}A_n$ and in case 2, the $\cup X_n=X$. Can someone enlighten me? Thank you.

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