Union of intersections and intersection of unions Would someone be able to help me understand the (concept of) "union of intersections" and the "intersection of unions".
More specifically, how to approach (prove) the following equality of two sets $X$ and $Y$;
$$
X=\bigcup_{n=1}^{\infty}\bigcap^\infty_{j=n} A_{j} \text{  and  } Y=\bigcap^\infty_{n=1}\bigcup_{j=n}^{\infty} A_{j} 
$$
Your feedback is much appreciated as always.

Suppose $x \in X$. Then $x \in \displaystyle\bigcap_{j=n}^\infty A_j $ for at least one $n = 1, 2, 3, \ldots$. Thus there exists a natural number $n = N$ such that $x \in A_j$ for every natural number $j \geq N$, which implies that $x \in \displaystyle\bigcup_{j=n}^\infty A_j$ for every $n = 1, 2, 3, \ldots$, and so $x \in Y$. Thus $X \subset Y$.
Conversely, suppose that $y \in Y$. Then $y \in \displaystyle\bigcup_{j = n}^\infty A_j$ for every $n = 1, 2, 3, \ldots$. Let $n \in \mathbb{N}$ be arbitrary. Then $y \in \displaystyle\bigcup_{j = n}^\infty A_j$, and so $y \in A_j$ for at least one natural number $j \geq n$. Thus for every natural number $n$, there exists a natural number $m \geq n$ such that $y \in A_m$.
As has been shown in the other answer. We cannot prove the inclusion $Y \subset X$ in general.
