Maximum principle for plurisubharmonic functions

Choose a small Euclidean ball $B(a,r)\subseteq \mathrm{\Omega }$. Fix $z\in B(a,r)$, and restrict $u$ to the affine complex line through $a$ and $z$. By [L1], that restriction is subharmonic on the connected line-domain component containing the segment from $a$ to $z$, and the global bound makes its value at $a$ a finite maximum. Hence [L2] makes it constant on that component, so $u(z)=u(a)$. Thus $u$ is constant on $B(a,r)$.

Put $S:=\{z\in \mathrm{\Omega }:u(z)=u(a)\}$. It is nonempty. Every point of $S$ is another point where the finite global maximum is attained, so the argument of step 1.1 makes $S$ open. Since $u\le u(a)$ everywhere and upper semicontinuity makes $\{u\ge u(a)\}$ closed, $S=\{u\ge u(a)\}$ is closed. Connectedness of $\mathrm{\Omega }$ gives $S=\mathrm{\Omega }$, so $u$ is constant.