Question: All vectors are real. Prove that $0 < x < y$ (element-wise) implies $\|x\| < \|y\|$ for any norm. This is probably very basic, but I don't seem to get the hang of it.

Edit: **it turns out this is not true.**

This is the first quadrant view of the unit ball of the norm suggested by @Daniel Fischer, which violates this.

![norm ball](https://i.stack.imgur.com/yFA7m.png)

Answer: 
In the setting of Banach lattices this is not true:

take $x=(1-\tfrac{1}{n})\_{n=1}^\infty, y = (1)\_{n=1}^\infty\in \ell\_\infty$. Then $x<y$ but $\|x\|=\|y\|=1$.
This holds in $\mathbb R^n$ equipped with the $L^p$ norm if $1 \leq p < \infty$, while this fails if $p = \infty$.