Analytic $F:\Omega\to\Omega'$ is proper iff $\{p_i\}\subset\Omega$ has no limit point in $\Omega$ implies $\{F(p_i)\}$ has no limit point in $\Omega'$

Question

I'm reading Rudin's Function Theory in the Unit ball of $\mathbb{C}^n$. I've just embarked on reading Chapter 15, but right out of the gate he makes an assertion that I cannot justify:

We shall study proper holomorphic maps $F:\Omega\to\Omega'$, where $\Omega$ and $\Omega'$ are regions in $\mathbb{C}^n$ and $\mathbb{C}^k$, respectively. In this context the compactness of $F^{-1}(K)$ for every compact $K\subset\Omega'$ is equivalent to the following requirement: If $\{p_i\}$ is a sequence in $\Omega$ that has no limit point in $\Omega$, then $\{F(p_i)\}$ has no limit point in $\Omega'$.

I don't know how to show this. The $\Rightarrow$ direction seems like it follows from the fact that proper maps are closed, but I don't have an idea how to show the converse. Any help is appreciated. Thanks.

Answer 1

Hint: If $(p_j)$ is a sequence in $\Omega$ then $(p_j)$ has no limit point in $\Omega$ is and only if for every compact $K\subset\Omega$ there exists $N$ so that $p_j\in\Omega\setminus K$ for all $j>N$.