I don't follow the argument in line 3-4 that reads "Since $U$ is an open set containing $x$, there would be a point $y$ in $U \cap A$." If this line is indeed logical, could someone please elaborate on it?

![enter image description here](https://i.stack.imgur.com/WL2Gi.jpg)
![enter image description here](https://i.stack.imgur.com/KdLp3.jpg)