Possible Duplicate:
Is norm non-decreasing in each variable?
Let $\| \cdot \|$ be any norm on $\mathbb{R}^{2}$. Let $0 \leq a \leq c$ and $0 \leq b \leq d$. Show that $\|(a,b)\| \leq \|(c,d)\|$.
Letting $A=(a,b)$ and $B=(c-a,d-b)$, this would follow if we could show $\|A-B\| \leq \|A+B\|$, as this would give
$2\|A+B\| \geq \|A-B\| + \|A+B\| \geq \|A-B+A+B\| = 2\|A\|$,
as desired. When $d=2b$, it is easy to show this using the following lemma I already proved: given $a,b,c,d$ as above,
$\|(a,0)\| \leq \|(c,0)\|$ and
$\|(0,b)\| \leq \|(0,d)\|$.
But I don't know how to prove this when $d \neq 2b$.