I know that a function $u \in C^2(\Omega)$ is subharmonic on $\Omega$ if $\Delta u(x) \ge 0$ for all $x \in \Omega$. But I have just acquainted with another definition of subharmonic functions which is given as follows $:$
"A function $u \in C^0 (\Omega)$ is subharmonic on $\Omega$ if for any $h \in C^2(\Omega) \cup C^0 (\bar \Omega)$ with $\Delta h = 0$ on $\Omega$ and for every ball $B \subset \subset \Omega$, $u \le h$ on $\partial B \implies u \le h$ on $B$."
To see that previous definition implies the later is immediate. For this we just need to apply Strong Maximum Principle on the function $w:=u-h$ by observing that $w \le 0$ on $\partial B$ and $h$ is superharmonic on $\Omega$ since it is harmonic on $\Omega$. But how can I do the converse part? Please give me some suggestions.
Thank you in advance.