Convergence of Complex Conjugate of Convergent Complex Sequence

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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:

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

$\blacksquare$