I want to prove that $$|z_1 z_2 z_3\cdots z_n|=|z_1| | z_2| |z_3|\cdots|z_n|$$
Now this really feels like something I can just throw induction at.
Base case: $n=2$.
$|z_1 z_2|=|z_1| |z_2|$ follows directly from expressing these numbers in exponential form.
Induction hypothesis:
Now assume that for $\forall k$, we can say that
$|z_1 z_2 z_3\cdots z_k|=|z_1| | z_2| |z_3|\cdots|z_k|$.
Inductive step:
Call the product of the first terms $w$
$$|z_1 z_2 z_3\cdots z_k z_{k+1}|=| w \space z_{k+1}|=|w||z_{k+1}|$$
Now use the induction hypothesis:
$$|w||z_{k+1}|=|z_1| | z_2| |z_3|\cdots|z_k||z_{k+1}|.$$
The aforementioned theorem holds by the principle of induction. QED
Modulus really feels like a homomorphism like this, did I do my proof correctly, should I elaborate? Is my writing clear? Any tips or recommendations? I really feel induction proofs are a bit rigid sometimes, the interesting ones are the ones where the inductive step isn't straightforward at all and requires some extra skill or inspiration.
Your solution is perfect. There is nothing to add. Perhaps only this for $n=2$:
$$ |zw| = \sqrt{zw \cdot \overline{zw}} = \sqrt{z\overline{z}}\cdot \sqrt{w\overline{w}} = |z||w|$$