Consider the sequence of sets $S(n)=\{1,2,3,\ldots,n\}$. It's common to write:
$$\bigcup_{k=1}^{∞}S(k)=N$$
Which I think is the same as:
$$\lim_{n\to \infty}\bigcup_{k=1}^{n}S(k)=N$$
Right? It doesn't make any difference if $k$ starts from $1$ or from any natural number $m$. What if we choose $m=n-1$?
$$\lim_{n\to \infty}\bigcup_{k=n-1}^{n}S(k)=N$$
Is it still true? Now why would we need the $S(n-1)$ when it's contained in $S(n)$. So It comes down to:
$$\lim_{n\to \infty}S(n)=N$$
Now it looks like a meaningless formula. How to make sense of this process?
The nested unions and intersections are widespreadly used. They just look unnecessary by the following process.
Your last formula is absolutely meaningful! In fact for any natural number $n$ we have$$n\in S(m)\qquad,\qquad m\ge n$$so we may conclude that$$\lim_{n\to \infty}S(n)$$contains all positive integers i.e. $$\lim_{n\to \infty}S(n)=\Bbb N$$