Definition $:$ Let $G \subseteq \mathbb C$ be a domain. A function $s : G \longrightarrow \mathbb R \cup \{-\infty\}$ is said to be subharmonic if
$(1)$ $s$ is upper semicontinuous.
$(2)$ given $D \subset \subset G$ and a continuous function $h : \overline D \longrightarrow \mathbb R$ with $h \rvert_{D}$ harmonic, if $s \leq h$ on $\partial D,$ then $s \leq h$ in $D.$
Problem $:$ Show that sum of two subharmonic functions is subharmonic.
Here I find one approach. But the answerer was himself confused about one of his arguments as to why there exist continuous functions $\alpha, \beta$ with $f \leq \alpha$ and $g \leq \beta$ such that $\alpha + \beta = u + \varepsilon.$ Since $f$ and $g$ are both upper semicontinuous we can get hold of decreasing sequences of continuous functions $\{\alpha_n\}_{n \geq 1}$ and $\{\beta_n\}_{n \geq 1}$ such that $\alpha_n \downarrow f$ and $\beta_n \downarrow g$ as $n \to \infty.$ But it is not quite clear to me as to why we can always get hold of a continuous function $\alpha$ such that $f \leq \alpha \leq u - g + \varepsilon$ using compactness of $\partial D.$ Any suggestion in this regard would be greatly appreciated.
Thanks for your time.