I tried proving that if $\lim_{n \to ∞} z_n=L_1$ then $\lim_{n \to ∞} \overline{z_n}=\overline{L_1}$ for $z_n \in \mathbb{C}$. This is my attemt.
Let $\epsilon>0$. Then there exists $N\in \mathbb{N}$ such than $|z_n-L_1|<\epsilon$ for each $n>N$.
Now let $n>N$. $|z_n-L_1|<\epsilon$ implies that $\overline{|z_n-L_1|}<\epsilon$.
Thus $|\overline{z_n}-\overline{L_1}|<\epsilon$.
I am not at all sure of this proof. Hope someone can assist me. Thanks