 Solution

Let $B⋐\Omega$ and let $h$ be the harmonic function whose boundary values are $\max (u_1,u_2)$. The harmonic extensions $(u_i{)}_B$ satisfy $(u_i{)}_B\le h$ by the maximum principle for harmonic functions, while subharmonicity gives $u_i\le (u_i{)}_B$ in $B$. Hence $\max (u_1,u_2)\le h$, proving that the maximum of two subharmonic functions is subharmonic.