Product of Complex Conjugates/General Result

Theorem
Let $z_1, z_2, \ldots, z_n \in \C$ be complex numbers.

Let $\overline z$ be the complex conjugate of the complex number $z$.

Then:
 * $\displaystyle \overline {\prod_{j \mathop = 1}^n z_j} = \prod_{j \mathop = 1}^n \overline {z_j}$

That is: the conjugate of the product equals the product of the conjugates.

Proof
Proof by induction:

For all $n \in \N_{> 0}$, let $\map P n$ be the proposition:
 * $\displaystyle \overline {\prod_{j \mathop = 1}^n z_j} = \prod_{j \mathop = 1}^n \overline {z_j}$

$\map P 1$ is trivially true, as this just says $\overline {z_1} = \overline {z_1}$.

Basis for the Induction
$\map P 2$ is the case:
 * $\overline {z_1 z_2} = \overline {z_1} \cdot \overline {z_2}$

which has been proved in Product of Complex Conjugates.

This is our basis for the induction.

Induction Hypothesis
Now we need to show that, if $\map P k$ is true, where $k \ge 2$, then it logically follows that $\map P {k + 1}$ is true.

So this is our induction hypothesis:
 * $\displaystyle \overline {\prod_{j \mathop = 1}^k z_j} = \prod_{j \mathop = 1}^k \overline {z_j}$

Then we need to show:
 * $\displaystyle \overline {\prod_{j \mathop = 1}^{k + 1} z_j} = \prod_{j \mathop = 1}^{k + 1} \overline {z_j}$

Induction Step
This is our induction step:

So $\map P k \implies \map P {k + 1}$ and the result follows by the Principle of Mathematical Induction.

Therefore:
 * $\displaystyle \forall n \in \N: \overline {\prod_{j \mathop = 1}^n z_j} = \prod_{j \mathop = 1}^n \overline {z_j}$