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:

1. $\Vert X \Vert \_p = \left ( \sum \limits\_{i = 1} ^{n} \vert x\_i
 \vert ^p \right ) ^{1/p}, p \in [1, + \infty)$
2. $\Vert X \Vert \_{\infty} = \max \limits\_{1 \le i \le n} \vert x\_i
 \vert $
3. In case $n = 1,$ $\Vert X \Vert = \Vert x \Vert = \Vert 1 \Vert \vert x \vert$