Does $1 < |z| < 2$ imply $|\frac{1}{z}| < 1$?
This is what I tried:
Divide given inequality by 2: $\frac{1}{2} < |\frac{z}{2}| < 1$
Apply reciprocal operation: ${2} > |\frac{2}{z}| > 1$
Divide by 2: $1 > |\frac{1}{z}| > \frac{1}{2}$
Hence, $|\frac{1}{z}| < 1$
Did I get that correct?
Yes, the conclusion is true. Visually, we can see:
The graph of $y=\left |\frac{1}{x} \right |$ is in purple, and the region where $1 < |x| < 2$ is shaded in red. We can see that the value of $\left |\frac{1}{x} \right |$ is certainly less than $1$ in the red region.