Can someone please point out the flaw in my proof?

Question

Let $f:X \to Y$ be a proper map.Show that $f$ takes discrete sets to discrete sets.

Proof:Let $A$ be discrete in $X$ and let $K$ be compact in $Y$ then $f(A) \cap K=f(A \cap f^{-1}(K))$,is finite since $A \cap f^{-1}(K)$ is finite.Hence $f(A)$ is discrete

But @Alex Ravsky here Does proper map $f$ take discrete sets to discrete sets? gave a counter example. So can Someone please point out the flaw in my proof?

Answer 1

$K$ does not necessarily equal $f(f^{-1}(K))$.