Just a hint of how I could prove this...

[Guest]

Prove that if z1z2 = 0 then z1 = 0 or z2 = 0 or both.

thanks in advance

 

[pka]

We are given \(\displaystyle z_1 \cdot z_2 = 0\). If \(\displaystyle z_2 = 0\) then there is nothing more to prove.

So suppose that \(\displaystyle z_2 \not= 0\), then from the field properties of real numbers \(\displaystyle \frac{1}{{z_2 }}\) the multiplicative inverse exists.
And
\(\displaystyle \begin{array}{l}
0 = z_1 z_2 \\
0\left( {\frac{1}{{z_2 }}} \right) = z_1 z_2 \left( {\frac{1}{{z_2 }}} \right) \\
0 = z_1 \\
\end{array}\),
we are done.