Does $||x||\leq ||\overline{x}||$ and $||y||\leq ||\overline{y}||$ imply $||x-y||\leq ||\overline{x}-\overline{y}||$?

Question

I've been working with norms for quite a bit now, and I have started to ponder whether

$$||x||\leq ||\overline{x}||\text{ and }||y||\leq ||\overline{y}||\Rightarrow||x-y||\leq ||\overline{x}-\overline{y}||?$$

I can't seem to be able to verify whether this is true or not so I was wondering if anyone can help me out?

Thanks in advance.

Answer 1

That's not even true for numbers: $|1|\leq|2|$, $|2|\leq|2|$, $|1-2|\not\leq|2-2|$.