I am trying to prove if $x,y,z \in \mathbb{R}/\mathbb{Z}$ with $x+y+z=0$ in $\mathbb{R}/\mathbb{Z}$, then \begin{align*} \|x\|+\|y\|+\|z\|=2\max(\|x\|,\|y\|,\|z\|) \end{align*} or
\begin{align*} \|x\|+\|y\|+\|z\|=1 \end{align*} with $\|x\|$ is the distance of $x$ to nearest integer.
I can only see $\|x\|+\|y\|+\|z\|=1$, and haven't found a way to get $\|x\|+\|y\|+\|z\|=2\max(\|x\|,\|y\|,\|z\|)$. Thank you.
