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I have a doubt.

What does it mean this notation $$ \bigcup_{n=1}^{\infty}A_n $$ where $A_n$ is a sequence of sets.

Is there a limit involved ?

Because, for instance, in the case of a series , we write the infinite summation $$ \sum_{n=1}^{\infty} a_n $$ But this is simply a notation to denote a series and its, eventual, limit. It should be written, more correctly, as $$ \sum_{n=1}^{\infty} a_n = \lim_\limits{n\rightarrow\infty} S_n $$ Where $S_n$ is defined as $\sum_{k=1}^{n} a_k$.

I have difficulties in understanding the nature of the quantities involved in the definition of superior/inferior limit.

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  • $\begingroup$ I understand that if $\{A_1, A_2, A_3,...\}$ is a countable collection of sets, then the union of all is $\cup_{n\in N} A_n$ or $\cup_{n=1} ^\infty A_n$. So, no limit in there. But the option that @JMoravitz pointed out is also possible. $\endgroup$ Mar 5, 2017 at 21:22
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    $\begingroup$ There is a bit of ambiguity as to what is truly meant by $\lim$ in this context since "distance between sets" isn't particularly well defined... Perhaps in this case it is easiest to just define it directly: $\bigcup\limits_{n=1}^\infty A_n := \{x~:~\exists n\in\Bbb N~\text{such that}~x\in A_n\}$. Similarly $\bigcap\limits_{n=1}^\infty A_n :=\{x~:~x\in A_n~\forall n\in\Bbb N\}$ $\endgroup$
    – JMoravitz
    Mar 5, 2017 at 21:28

2 Answers 2

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$A=\bigcup_{n=1}^{\infty}A_n$ means $x\in A$ iff there exists $k\in\Bbb{N}$ such that $x\in A_k$. So there is no notion of limit really needed. In order to define limits you need more structure on your sets (an order for example) but it not strictly necessary here.

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Supplement to the answer by Maczinga and to some comments: You will also see $\cup F$ when $F$ is a set of sets. $$x\in \cup F \iff \exists f\in F\ :x\in f$$

For example if $F=\{A_n:n\in \mathbb N\}$ then $\cup F$ and $\cup_{n=1}^{\infty}A_n$ are the same thing.

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