Whether this vector norm proposition is true? $ \vert x_i \vert \ge \vert y_i \vert \Rightarrow \Vert X \Vert \ge \Vert Y \Vert$ Whether this vector norm proposition is true? 
$ \vert x_i \vert \ge \vert y_i \vert \Rightarrow \Vert X \Vert \ge \Vert Y \Vert$
where $X, Y \in \mathbb R ^n$
Is it true for all kinds of norms?
I've attempted some norms below and it is obviously true for them:


*

*$\Vert X \Vert _p = \left ( \sum \limits_{i = 1} ^{n} \vert x_i
    \vert ^p \right ) ^{1/p}, p \in [1, + \infty)$ 

*$\Vert X \Vert _{\infty} = \max \limits_{1 \le i \le n} \vert x_i
    \vert $ 

*In case $n = 1,$ $\Vert X \Vert = \Vert x \Vert = \Vert 1 \Vert \vert x \vert$

Define a norm on $\Bbb R^2$ by $$||(x,y)||=|x|+|x-y|.$$
