understanding algebraic proof in complex geometry

Question

I am studying "Condensed Mathematics and Complex Geometry" by Dustin Clausen, Peter Scholze. I came across this theorem and this proof:

I don't understand a lot of steps in the proof. I proved a few steps, including that the kernel of $\vert \cdot \vert$ is an ideal, among other things. However I have missed the whole point of the proof, my two biggest issues being :

1. (Around the red dot in the image), it says "Our goal in then to show that $A=\mathbb{C}$. Does that mean that A is isomorphic to $\mathbb{C}$? If it is the case, how was this achieved?
2. There isn't much talk of continuity except for the function on the 6th line of the proof. Where, then did they prove anything about a homeomorphism?