(related to Corollary: Convergence of Complex Conjugate Sequence)
According to the corresponding proposition \((c_n)_{n\in\mathbb N}\) is a convergent complex sequence with \[\lim_{n\to\infty}c_n=c\]
if and only if
\[c=a+ib,\quad a=\lim_{n\to\infty}\Re(c_n),\quad b=\lim_{n\to\infty}\Im(c_n).\]
Because
\[\overline{c}=a-ib,\quad a=\lim_{n\to\infty}\Re(\overline{c_n}),\quad -b=\lim_{n\to\infty}\Im(\overline{c_n}),\]
it follows \[\lim_{n\to\infty}\overline{c_n}=\overline{c}.\]
