Convergence of Complex Conjugate of Convergent Complex Sequence

Theorem
Let $z \in \C$.

Let $\sequence {z_n}_{n \mathop \in \N}$ be a complex sequence converging to $z$.

Then:


 * $\overline {z_n} \to \overline z$

Proof
Let $\epsilon > 0$.

Since $z_n \to z$, from the definition of convergence, we can find $N \in \N$ such that:


 * $\cmod {z_n - z} < \epsilon$

From Complex Modulus equals Complex Modulus of Conjugate, we have:


 * $\cmod {\overline {z_n - z} } = \cmod {z_n - z}$

From Difference of Complex Conjugates, we have:


 * $\cmod {z_n - z} = \cmod {\overline {z_n} - \overline z}$

So we have:


 * $\cmod {\overline {z_n} - \overline z} < \epsilon$

for each $n \ge N$.

Since $\epsilon$ was arbitrary, we have:


 * $\overline {z_n} \to \overline z$