Question 940979
Note that *[tex \large |(1+i)z^3 + iz| \le |(1+i)z^3| + |iz|] by the triangle inequality.


Since *[tex \large |z| < 0.5], it follows that *[tex \large |iz| < 0.5] and *[tex \large |z^3| < \frac{1}{8}], and since *[tex \large |1+i| = \sqrt{2}], we have *[tex \large |(1+i)z^3| < \frac{\sqrt{2}}{8}]. Combining,


*[tex \large |(1+i)z^3 + iz| \le |(1+i)z^3| + |iz| < \frac{\sqrt{2}}{8} + 0.5 < \frac{3}{4}]