# A problem with the concept of limit in sets

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.

• The problem is that you chose $m$ depending on $n$. That makes the difference. Oct 11, 2018 at 18:41
• All of the statements you make are true for this specific situation. Consider a different example where instead $T(n) = \{n\}$. Here you have $\bigcup\limits_{k=1}^\infty T(n)=\Bbb N$ as well and you have $\bigcup\limits_{k=1}^n T(n) = S(n)$, but here you actually don't have any useful meaning you can give to $\lim\limits_{n\to\infty} T(n)$. You could give a useful meaning to $\lim\limits_{n\to\infty} S(n)$ however. Oct 11, 2018 at 18:42
• Read about Set-theoretic limits on wikipedia. Oct 11, 2018 at 18:43
• @JMoravitz, I've had a look on that. The problem is that this format is used often in Stein's real analysis without those liminf and limsup stuff. Thanks for your insight though. Oct 11, 2018 at 18:48

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. $$\lim_{n\to \infty}S(n)=\Bbb N$$