Let $x=[x_1,\dots,x_n],y=[y_1,\dots,y_n]\in\mathbb{R}^n$. Suppose the elements of $x$ and $y$ are nonnegative and satisfy $$0\le x_i\le y_i$$ for $i=1,\dots,n$.
The question is: do we have $\|x\|\le\|y\|$ for all norms (not just $\ell_p$ norms)?
It is trivial to prove for $\ell_p$ norms. However, how to prove it for all norms based on the fundamental properties of norms: (i) $\|x\|\ge0$ (ii) $\|ax\|=|a|\|x\|$, (iii) $\|x+y\|\le\|x\|+\|y\|$
No. Take the norm $\|\cdot\|$ on $\mathbb{R}^2$ given by $$\|(a,b)\|=\max \{|a|, |b-a|\}.$$ (One has to check that this is a norm). Then the vector $(1,2)$ has norm $1$, because $$\|(1,2)\|=\max\{1, 2-1\}=1.$$ However, the vector $(0,2)$, which is pointwise between $(0,0)$ and $(1,2)$, has norm $2$.