Answer
See below.
Work Step by Step
Let $z_1=a+bi$ and $z_2=c+di$. Then $\overline{z_1+z_2}=\overline{(a+bi)+(c+di)}=\overline{(a+c)+(b+d)i}=(a+c)-(b+d)i$
$\overline{z_1}+\overline{z_2}=\overline{(a+bi)}+\overline{(c+di)}=a-bi+c-di=(a+c)-(b+d)i$. Thus they are equal, thus we proved what we wanted to.