Complex Analysis proof problem I have an feeling this bad proof because I am assuming my conclusion and I go into repeating. This isn't allowed in mathematics, so I need help to refix my steps.
So the proof I am proving is |ab| = |a||b|
So I started with:
Let a,b $\in \mathbb{C}$
I claimed $|ab|^{2} = |a|^{2} |b|^{2}$ is true
so I went to from LHS = RHS (to say LHS is equal to RHS) by using modulus of complex which is $|z|^{2}$ = $zz^{*}$
$|ab|^{2} = aa^{*} (bb^{*})$ = $(ab)(ab)^{*}$ = $|ab|^{2}$
then i took the square root of $|ab|^{2} = |a|^{2} |b|^{2}$ to give me |ab| = |a||b| because modulus is never negative, and we only take the positive.
this is where I believe this proof is 100% written wrong because I am assuming my claim is true (unproven to be exact) and it repeats again. How do I fix this?

$|ab|^{2}=(ab)(ab)^{*}=aba^{*}b^{*}=(aa^{*}) (bb^{*})=|a|^{2}|b|^{2}$. Now take square root.
