Finding the minimum value of a complex number If $z$ is a complex number satisfying $|z^2+1| = 4|z|$ . Then prove that the minimum value of $|z|$ is $4$
This is how I attempted the problem ,
$\frac{|z^2+1|}{|z|} = 4$
Therefore , 
$|z + \frac{1}{z}| = 4$
How do I proceed from here ? 
According to the solution of the above problem the next step would be as follows 
$||z| - |\frac{1}{z}|| ≤ 4$
We can then form a quadratic in $z$ to find out the least value of $z$. However , I’m stuck at the first step of the solution . How does $|z + \frac{1}{z}| = 4$ imply $||z| - |\frac{1}{z}|| ≤ 4$ ? Please help .

The implication$$\left|z+\frac1z\right|=4\implies\left||z|-\left|\frac1z\right|\right|\leqslant4$$comes from the inequality$$(\forall z,w\in\mathbb{C}):|z-w|\geqslant\bigl||z|-|w|\bigr|.$$
