elementary set theory - Question regarding the union set of the family and intersection set of the family

I'm really struggling with this question.

"Let $\{A_n\}_{n\in\mathbb N}$ be a family of subsets of a set $S$. Let
$$X:=\bigcup_{n\in\mathbb N}\left(\bigcap_{k\geq n}A_k\right),\qquad Y:=\bigcap_{n\in\mathbb N}\left(\bigcup_{k\geq n}A_k\right).$$
Does any of the relations X ⊂ Y, X = Y, Y ⊂ X hold?"

Currently, by using the definition of Unions and Intersections I already proved that X ⊂ Y. However I'm stuck trying to prove whether the inverse, Y ⊂ X holds. Intuitively, something tells me that Y ⊂ X holds, but I can't prove it correctly. I would appreciate some help. Thanks.

Answer

Suppose $A_{2k}=\{1\}$, and $A_{2k+1}=\{2\}$, for all $k\ge 0$; i.e. the even-indexed sets are all the same, and the odd-indexed sets are also all the same. You will find that $X=\emptyset$ while $Y=\{1,2\}$.