Is every norm increasing?
The answer is no.
Consider the norm $\|(x,y)\| = |x| + |x-y|$ on $\mathbb{R}^2$.
We have $(1,0) \le (1,1)$ in the sense you defined but
$$\|(1,0)\| = 2, \quad \|(1,1)\| = 1$$
There is a nontrivial theorem characterizing this property:
Let $\|\cdot\|$ be a norm on $\mathbb{C}^n$. The following is equivalent:
- $|x_i| \le |y_i|, \forall i=1, \ldots, n$ implies that $\|(x_1, \ldots, x_n)\| \le \|(y_1, \ldots, y_n)\|$.
- $\|(x_1, \ldots, x_n)\| = \|(\left|x_1\right|, \ldots, \left|x_n\right|)\|, \forall (x_1, \ldots, x_n) \in \mathbb{C}^n$.