If $\lim\limits_{z \to \infty} |F(z)|=0$, can $\lim\limits_{z \to \infty} F(z) \ne 0$?

Question

To prove the existence of limit of $F(z)$ at $z=\infty$ in here (Eq. 1.8 page 3) it says that since $\lim\limits_{z \to \infty} |F(z)|=0$ thus $\lim\limits_{z \to \infty} F(z)=w_n$ exists; Of course exits, but also must $w_n=0$. But Nor this article neither the book Churchill Sec 117 Exercise 2 doesn't say $w_n=0$. Can $w_n \ne 0$ be possible?

Answer 1

The limit must be zero. Suppose $\lim F(z)=w_n$ exists. Then $\lim | F(z)-w_n |=0$. However $|F(z)-w_n| \geq \big||F(z)|-|w_n| \big|$, for any $z$ by the triangle inequality, so it follows that $\lim \big||F(z)|-|w_n| \big|=0$. That is, $\lim |F(z)|=|w_n|$.

Note that the convergence of $|F(z)|$ does not imply $F(z)$ converges unless $\lim |F(z)|=0$.