I am struggling to prove the following complex inequality:
$\left | z+4 \right |-2\leq \left | z+2 \right |, \text {where} \:z \in \mathbb{C} $
My thought process so far is to use the triangle inequality for complex numbers:
$\\\left | z_{1}+z_{2} \right |\leq \left | z_{1} \right |+\left | z_{2} \right |, \text {where} \:z_{1}, z_{2} \in \mathbb{C}$
And that gives:
$\\\left | z + 4 \right |\leq \left | z \right |+\ 4\\\left | z + 4 \right |-2\leq \left | z \right | +2\geq \left | z+2 \right |$
But i seem to be stuck here and help would be appreciated
Observe that: $$|z+4| = |(z+2)+2| \leq |z+2|+2$$
by the triangle inequality.