Let me try again. Suppose $\|\cdot\|$ is a norm in $\mathbb{R}^n$ and let $f(x_1,...,x_n)=\|(x_1,...,x_n)\|$
where $x_i\geq 0, \forall i$. I want to prove or disprove that $f$ is an nondecreasing function in each of its variables.
Thanks
Note: Suppose we vary $x_i$ and fix the other variables. Then I want the function $g(x_i)=f((x_1,...,x_i,...,x_n))$ to be nondecreasing.