Does an inequality stay true after rooting

Question

I have a very simple question but can't seem to find the answer for it.
If I have an equation like so:

$\|z-w\|^2 \ge (\|z\|-\|w\|)^2$, where $z$ and $w$ are complex numbers and $\|\|$ is the modulus

Would the inequality (bigger or equal) stay true if I root them both? So would

$\|z-w\| \ge \|z\|-\|w\|$ still hold?

Answer 1

It is true that $$0\leq a \leq b$$ implies $$\sqrt a\leq \sqrt b.$$ In your case you should be a little bit more careful as $\sqrt{(\|z\|-\|w\|)^2}=|\|z\|-\|w\||$, which can be $\|z\|-\|w\|$ or $\|w\|-\|z\|$. But since absolute value is the bigger of the two, your inequality holds anyway.

Answer 2

In general, $a^2\ge b^2$ does not imply $a\ge b$. However, if $a\ge0$, it does. This is the case here.