I'll first express my confusion：why can the exhaustion function of a domain in $\mathbb{C}^n$ be bounded?
More precisely:
In p. 45 of the book Partial differential equations in several complex variables by Chen S C and Shaw M C. , the author give the definition of the exhaustion function as follows:
Definition 3.4.6. A function $\varphi: D \rightarrow \mathbb{R}$ on an open subset $D$ in $\mathbb{R}^n$ is called an exhaustion function for $D$ if for every $c \in \mathbb{R}$ the set $\{x \in D \mid \varphi(x)<c\}$ is relatively compact.
Then one can easily see from the definition of relatively compactness that when the point tends to the boundary of $D$, $\varphi$ tends to $+\infty$.
However, in p. 49, the author give a well-known theorem that there always exists a bounded strictly plurisubharmonic exhaustion function on any smooth bounded pseudoconvex domain:
Theorem 3.4.12. Let $D \subset \mathbb{C}^n, n \geq 2$, be a smooth bounded pseudoconvex domain. Let $r$ be a smooth defining function for $D$. Then there exist constants $K>0$ and $0<\eta_0<1$, such that for any $\eta$ with $0<\eta \leq \eta_0, \rho=-\left(-r e^{-K|z|^2}\right)^\eta$ is a smooth bounded strictly plurisubharmonic exhaustion function on $D$.
I restate my confusion：why can the exhaustion function of a
domain in $\mathbb{C}^n$ be bounded? Did I misunderstand something
somewhere? (p.s. I am very sure that Theorem 3.4.12. is right and
well-known)
Feel free to share your thoughts – all input are welcome!