Unpacking notation with $\inf$, $\sup$ as part of proof of open set $\mathbb{R}$ can be written as countable union of disjoint open intervals Here is a Proposition from my real analysis book.

Proposition. Suppose $G \subset \mathbb{R}$ is open. Then $G$ can be written as the countable union of disjoint open intervals.

The proof begins with the following.

Let $G$ be an open subset of the real line and for each $x \in G$, let$$A_x = \inf\{a : \text{there exists }b\text{ such that }x \in (a, b) \subset G\}$$and$$B_x = \sup\{d : \text{there exists }c \text{ such that }x \in (c, d) \subset G\}.$$Let $I_x = (A_x, B_x)$.

I just realized I don't really get what this beginning part is saying. Can somebody explain it to me/add some more detail? Thanks in advance!

I think it might be easier to understand if it were written as
$$A_x = \inf\{ a : (a,x) \subset G\}$$
$$B_x = \sup\{ d : (x,d) \subset G\}$$
Basically, $A_x$ is the left endpoint of the largest open interval containing $x$ and $B_x$ is the corresponding right endpoint.
