We were given this hint . (Hint: $||z| − |w|| ≤ |z − w|$ for all
$z, w ∈ C.$)

From what I understood in lecture, it's some epsilon delta proof? I don't understand how to prove continuity at all using this method or any other and I'm really lost): please be very explicit with your help if you can because i really suck at this and I've been staring at this question for hours