making sense of a very basic inequality proof in an intro analysis textbook This is an excerpt from an Analysis textbook by Jospeh Taylor.
proof from the book
The author gets to line (2.1.1) and writes "If we interchange ...... also holds." I don't understand this step in the proof. Why is the author adding an additional $|b|$ to both sides? Can you explain this section of the proof?
Why can't the author just write something like: $|a|-|b| ≤ |a − b|$ holds for all real $a,b$ and $|a-b|=|b-a|$, therefore $|b|-|a| ≤ |a − b|$. Multiplying by $-1$ we get: $-|a − b| ≤ |a|-|b|$. Finally, $-|a − b| ≤ |a|-|b| ≤ |a − b|$ and therefore $||a|-|b|| ≤ |a − b|$. Note that the very last step is already proven in the text.
can someone help me make sense of the author's argument?

Since $|a-b|\geq|a|-|b$, we obtain $|b-a|\geq|b|-|a|$, which says $$|a-b|\geq|a|-|b|$$ and $$|a-b|\geq-(|a|-|b|),$$ which gives $$|a-b|\geq||a|-|b||$$
We used the following.

If $x\geq y$ and $x\geq-y$ so $x\geq|y|.$

This statement you proved in your post: Because it's $-x\leq y\leq x$.
Also, $$|a-b|+|b|\geq|b|-|a|$$ it's typo.
It should be $$|a-b|\geq|b|-|a|.$$
