I'm reading the proof of the uniformization theorem from a pdf by Joel W. Robbin. On page 5, there's a corollary about subharmonic functions. My question is about proposition (iv):
"If $u: X \to \mathbb{R}$ is continuous, positive and harmonic on an open set $V$, and vanishes on $X \setminus V$, then $u$ is subharmonic on $X$."
But I think the proof has a problem in its last statement. The maximum principle states that a "nonconstant" subharmonic function doesn't assume its maximum but in the last statement of the proof, this case is ignored.
Is this proof correct? Or at least, Is this proposition correct?