Closed set of $\mathbb{C}$

Question

Let $\mu\in \mathbb{C}$. Put $U=\{z\in \mathbb{C}, Re(z)>0\}$.

Now, we define de set $F_\mu=\{a\in U, \mu\in\{na,n\in N^* \} \}$.

I want to show that the set $F_\mu$ is closed in $\mathbb{C}$.

If $(a_p)\subset F_\mu$ such that $a_p\to a$.

Then for all $p\in N$, there exists $ n_{p}\in N$ such $\mu= n_{p}a_p$.

My question how i can to prove that the set is closed?

Answer 1

Let $\mu=1$. Then $F_1=\{\frac{1}{n}: n \in \mathbb N\}$. $F_1$ is not closed.