I am self studying from the book of Rick Miranda "Algebraic Curves and Riemann Surfaces."
On page 41, the end of proof of Proposition 3.12 finishes with the statement:
Discrete subsets of compact spaces are finite
However, this is generally not true, as this shows. So is the proof wrong, or have I failed to take in account something, like for example that we are working with Riemann surfaces?
If the proof is wrong, is there any way to fix it?