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

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.

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

• Simple and concise(+1)! – Chinnapparaj R Apr 20 at 9:16

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 $$b=-1$$?

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 $$|z|-1 \leq ||z|-|-1||\leq |z -(-1)| = |z+1|$$