Show that $f(x) = (\cos x, \sin x) $ is not a closed map using a specific example

Question

In an accepted answer on showing that $f(x) = (\cos x, \sin x) $ is not a closed map for $x \in \Bbb R$, the author considers the example $F= \{2\pi n + 1/n, n=1,2,3,...\} $ which is a closed set. I am not sure why the $f(F)$ is not a closed set. As far as I can see, $f(F)$ is an infinite set of isolated points around $(1,0)$. Any insights appreciated.

Answer 1

$(1,0)$ is an element of the closure of $f(F)$, since it is the limit as $n\to\infty$ of $f(2\pi n+1/n)$, yet $(1,0)\notin f(F)$.