Measure, Integration & Real Analysis · Problem 6B.3

Banach Spaces — Sheldon Axler

Mathematics — Measure Theory · complex-numbers-absolute-value-bounds-2

Problem

Suppose

z \in \mathbf{C}

. Prove that

\dfrac{|\operatorname{Re} z| + |\operatorname{Im} z|}{\sqrt{2}} \leq |z| \leq |\operatorname{Re} z| + |\operatorname{Im} z|