This is Exercise EP $14$ from Fernandez and Bernardes's book Introdução às Funções de uma Variável Complexa (in Portuguese). The authors ask us to prove that the inequality $\sqrt{2}|z|\geq |\operatorname{Re}z|+|\operatorname{Im}z|$ holds for all $z\in \mathbb{C}.$
I know that $|z|\geq\max\{|\operatorname{Re}z|,|\operatorname{Im}z|\}$, but all I've managed to do is show the obvious: $2|z|\geq |\operatorname{Re}z|+|\operatorname{Im}z|.$
I would appreciate a hint here.