Could you please help me verify my proof for this very basic result in vector algebra? Are these steps, mathematically correct or is there a better way to prove this result?
Using the triangle inequality for vectors -
$$|u+v| \le |u| + |v|$$
prove that :
$$|a-b| \ge ||a|-|b||$$
Proof.
We have $|u+v|\le|u|+|v|$.
(1) Let $u=a-b$ and $v=b$. Then,
$|a| \le |a-b| + |b|$
$|a-b| \ge |a| - |b|$
(2) Next, let $v = b-a$ and $u = a$. Then,
$|b| \le |a| + |b-a|$
$|b| - |a| \le |a-b|$
$|a-b| \ge |b| - |a| = -(|a|-|b|)$
Combining (1) and (2), we must have :
$$|a-b| \ge ||a| - |b||$$
The proof is correct, but you should get used to put $\implies$ and $\iff$ whenever it is need. For instance: after putting $u=a-b$ and $v=b$, it would have been better if you had written that $|a|\leqslant|a-b|+|b|$ and that$$|a|\leqslant|a-b|+|b|\iff|a-b|\geqslant|a|-|b|.$$