I am reading subharmonic functions and their properties from the book *From Holomorphic Functions to Complex Manifolds* by Grauert and Fritzsche. Let me first define what a subharmonic function is.

According to the book I mentioned, a function $s : G \longrightarrow \mathbb R \cup \{-\infty\}$ (where $G \subseteq \mathbb C$ is a domain) is said to be subharmonic if the following conditions hold $:$

$(1)$ $s$ is upper semicontinuous.

$(2)$ For every disk $D \subset \subset G$ (i.e. $\overline D \subseteq G$ is compact) if there exists a continuous function $h : \overline D \longrightarrow \mathbb R$ with $h \lvert_{D}$ harmonic and $s \leq h$ on $\partial D,$ then $s \leq h$ on $D.$ This property is known as the harmonic majorant property.

An interesting fact about subharmonic functions is that they also satisfy maximum principle as in the case of harmonic functions. The proof of this given in the book is the following $:$

The proof is quite clear and understandable modulo the statement highlighted in yellow. I have searched many properties of upper semicontinuous functions in google but I didn't able to find the property highlighted in yellow. Could anyone have any insight on how does the claim work? Any suggestion would be warmly appreciated.

Thanks for your time.