Need clarification on notation for collection of subsets

Question

I would like to know the meaning of the following:
I need to prove that $\bigcup_{n=1}^{\infty}{A_n}=A$ is true, where A is a set of positive real numbers, and $A_n = [x\space\epsilon\space A: x\geq 1/n]$, where $n\space\epsilon\space\Bbb{N}$. I have two questions. Firstly, does this mean that $A_n$ can contain the same number multiple times? For example, could $A_4=[1/3,1/3]$? Secondly, and most importantly, what does the left hand side of the equality mean? I get the impression that it's the union of all the $A_n$ sets for every natural numbers, but I can't tell.
Thanks in advance.

Answer 1

Going off @Bungo's comment:

I usually try to avoid using "no so clearly defined" concepts in my proofs, like $A_{\infty}$. To me it just looks like a change in notation, i.e. you're putting

$$A_{\infty} := \bigcup_{n=1}^{\infty}{A_n}$$

which is clearly true since it's a definition. Now the question is whether $A_{\infty} = \mathbb{R}_+$.

Instead we could just prove the two inclusions $\bigcup_{n=1}^{\infty}{A_n} \subset \mathbb{R}_+ $ and $\mathbb{R}_+ \subset \bigcup_{n=1}^{\infty}{A_n} $

If $ x \in \bigcup_{n=1}^{\infty}{A_n}$ then for some $n$, $x \in A_n$. But each $A_n$ contains only positive real numbers, so $x$ must be one too, so $x \in \mathbb{R}_+$

And if $x \in \mathbb{R}_+$, then there is some $n_0$ s.t. $x \geq \frac{1}{n_0}$, so $x \in A_{n_0}$ by definition, and so $ x \in \bigcup_{n=1}^{\infty}{A_n}$.

So the two sets are equal, since they contain the same elements.