Let $E = \{E_k\}_{k \in \mathbb{N}}$ be an infinite sequence of sets. Then, the following inclusion holds:
$\bigcup_{n=1}^{\infty} \bigcap_{k=n}^{\infty} E_k \quad\subseteq\quad \bigcap_{n=1}^{\infty} \bigcup_{k=n}^{\infty} E_k$
I know the left-hand side (LHS) represents elements which belong to all but finitely many sets in the sequence $E$, and the right-hand side (RHS) represents elements which belong to infinitely many sets in the sequence $E$. This concludes the proof that the LHS is contained in the RHS. Yet it is not intuitive for me, since the above interpretation is not intuitive per se.
Is there an intuitive reason why the above inclusion holds? That is, (very informally) why union of intersections is contained in intersection of unions?