If someone could help me with this question I would really appreciate it.For some reason I am getting a weaker version of these inequalities when applying triangle inequality.
Let S be the interior of the circle |z − 1 − i| = 1. that if z ∈ S then √5 − 1 < |z − 3| < √5 + 1. Obtain the result geometrically by considering the line containing the center of the circle and the point 3.
Using triangle inequality: If $z \in S$ then $|z - 3| = |z - 1 - i + i - 2| < |z - 1 - i| + |i - 2| < 1 + \sqrt{5}$
$|z - 3| > |i - 2| - |z - 1 - i| > \sqrt{5} - 1$