Prove that $\displaystyle |\mathrm{Re}(z)|+|\mathrm{Im}(z)|\leq\sqrt{2}\|z\|$ for all $\displaystyle z\in\mathbb{C}$.
Not very hard, but I wonder to see if there will be nice solutions.
My own solution was also similar.
Just set $\displaystyle f(t):=|t|+|\sqrt{1-t^{2}}|$ for $\displaystyle t\in[-1,1]$, and show that $\displaystyle f(t)\leq f(\sqrt{2}/2)=\sqrt{2}$ for all $\displaystyle t\in[0,1]$.
Since $\displaystyle f$ is even, we have $\displaystyle f(\cos(t))\leq\sqrt{2}$ for all $\displaystyle t\in[0,2\pi]$.