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this question is different to this post, which introduces $\{0\}$ (my own mistake) that is not a member of $A_j$, and someone had posted an answer based on that, so it is not appropriate to modify the whole which would disable that answer.

I am learning this wiki page, which uses sequence of sets in the definition

Suppose that ${\displaystyle \{A_{n}\}_{n=1}^{\infty }}$ is a sequence of sets. The two equivalent definitions are as follows.

Using union and intersection, define

$\liminf_{n \rightarrow \infty} A_n = \bigcup_{n \ge 1} \bigcap_{j \geq n} A_j$

...

The sequence ${A_n}$ is said to be nondecreasing if each $A_n ⊂ A_{n+1}$

the simplest example of a (monotonic increasing) sequence I can imagine is the Natural number $(1, 2, … , n), \quad where \quad n \in \mathbb {N}^*$

I assume this $(\{1\},\{1,2\},...,\{1,2,…,n\}), \quad where \quad n \in \mathbb {N}^*$, is an nondecreasing sequence of sets.

to be clear, $A_1 = \{1\}, A_2 = \{1,2\}, ..., A_n = \{1,2,…,n\}$

limit infimum is defined as

$\liminf_{n \rightarrow \infty} A_n = \bigcup_{n \ge 1} \bigcap_{j \geq n} A_j$

to understand this easily, I would like to consider $\bigcap_{j \geq 1} A_j$ first

so, is $\bigcap_{j \geq 1} A_j$ equal to

$\{\{1\}\}$

or

$\{1\}$

I think it is the last one, and I need a double-check

Asaf Karagila
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JJJohn
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  • It is kind of confusing to write ${A_n}_{n=1}^{\infty}$ as notation of a sequence of sets. One can interpreted that as the sequence $A_1,A_2,\dots$ but also as the sequence ${A_1},{A_2},\dots$. Which one do you mean? And secondly for certainty: do we have $A_n={1,2,\dots,n}$ here? – drhab Aug 01 '19 at 15:19
  • @drhab I didn't mean that we have $A_n={1,2,\dots,n}$, if there is any part of my OP misleading that, please point it out, thanks in advance. – JJJohn Aug 01 '19 at 15:24
  • Then how is $A_n$ defined in your question? – drhab Aug 01 '19 at 15:27
  • I agree with you about the confusing sequence notation, which has been buzzing me a lot. I am writing another post about that. – JJJohn Aug 01 '19 at 15:28
  • @drhab $A_1 = {1}, A_2 = {1,2}, ..., A_n = {1,2,…,n}$ – JJJohn Aug 01 '19 at 15:35
  • So when you write ${A_n}{n=1}^\infty$, you really mean $(A_n){n=1}^\infty$? – TonyK Aug 01 '19 at 15:38
  • If you make it something like: "$A_1,A_2,\dots$ is a sequence of sets and $A_n={1,2,\dots,n}$..." then everything is clear to everyone. Also you can use $(A_n)_n$ instead of $A_1,A_2,\dots$. – drhab Aug 01 '19 at 15:38
  • And when you write $({1},{1,2},…,{1,2,…,n})$, you really mean $({1},{1,2},…,{1,2,…,n},\ldots)$? – TonyK Aug 01 '19 at 15:39
  • @TonyK Actually I am not sure if I should put ... at the end, would you please provide some solid reasons to do that on this thread notation of a sequence of sets – JJJohn Aug 01 '19 at 16:01
  • @drhab thanks man. Your notation is more clear. would you please mv that to the answer for this thread notation of a sequence of sets – JJJohn Aug 01 '19 at 16:03

1 Answers1

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Here $A_n= \{\{1\}, \{1,2\},\ldots,\{1,\ldots,n\}\}$. All sets $A_n$ have the element $\{1\}$ in common. So the intersection $\bigcap_{n\geq 1} A_n$ consists of the set $\{\{1\}\}$.

Haha, sorry, but in your case $A_n=\{1,\ldots,n\}$, $n\geq 1$, the intersection is indeed $\{1\}$.

I hope this distinction helps.

Wuestenfux
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  • I meant $A_1 = {1}, A_2 = {1,2}, ..., A_n = {1,2,…,n}$, my previous is not clear, sorry for that. – JJJohn Aug 01 '19 at 15:35