Meaning of Infinite Union/Intersection of sets

Question

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.

Answer 1

$A=\bigcup_{n=1}^{\infty}A_n$ 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.

Answer 2

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.