Question

Reading Demailly's Algebraic and Complex geometry in particular chapter 3 about positive currents, tha author defines the unbounded locus $L(u)$ of a plurisubharmonic function $u$ to be the set of points $x\in X$ such that $u$ is unbounded in every neighborhood of $x$. Then he shows that $L(u)$ contains the closure of $P(u)=u^{1}(-\infty)$, but in general this inclusion is strict: in fact, $u(z)=\sum k^{-2}\log(|z-1/k|+e^{-k^{3}})$ is everywhere finite in $\mathbb{C}$ but $L(u)=(0)$. what i don't understand is why is everywhere finite in $\mathbb{C}$, since if $z=1/k$ then $u(z)=-\infty$