Is the statement true?$|z+1|\ge|z|-1\ \forall z\in\Bbb{C}$

Question

I have tried to prove it in the following manner-
$||z|-1|=||z|-|1||\le|z-1|\ \forall z\in\Bbb{C}$ (by Triangle inequality)
Now, can I write $|z-1|\le|z+1|$ in $\Bbb{C}$? If yes then the proof is done.
But I can't get it. I don't know whether the statement is true. Can anybody solve it? Thanks for the assistance in advance.

Answer 1

$|z|=|z+1-1|\leq |z+1|+1$. Now just pull 1 on RHS to LHS.

Answer 2

The triangle inequality doesn't say that $||z|-|1||\le|z-1|$. It says that $|a+b|\le|a|+|b|$ for all $a,b$.

What does it say if $a=z+1$ and $b=-1$?

Answer 3

Your approach with the reverse triangle inequality ($||z|-|w||\leq |z-w|$) works well with a slight adjustment.

Just note that

• $a\leq |a|$ for all $a \in \mathbb{R}$ and
• letting $w = -1$ you get immediately $$|z|-1 \leq ||z|-|-1||\leq |z -(-1)| = |z+1|$$