Here are my "proofs" for Spivak's Calculus Chapter 1 Problem 12. I am new to this level of rigour and I am attempting to intimate myself with more advanced topics of mathematics to prepare for next year. I apologize in advance, as these so-called "proofs" are not likely to be nearly as rigorous as they should be. Any assistance on how to write the proofs better or any critiques on faulty logic would be greatly appreciated.
12) i) Prove that |xy|=|x|$\cdot$|y|
My proof is:
$x,y\in\mathbb{R} \therefore$ $|x|>0$ and $|y|>0$ $\therefore$ $|x|\cdot|y|>0$
${(|xy|)}^2 = (xy)^2 = x^2y^2 = {|x|}^2{|y|}^2 = (|x|\cdot|y|)^2$
Therefore, because $|x|\cdot|y|>0$, $\;$ $|xy|=|x|\cdot|y|$
12) ii) Prove that $\left|\frac{1}{x}\right|=\frac{1}{\left|x\right|}$
My proof is:
$\left|\frac{1}{x}\right|=|1|\cdot\left|x^{-1}\right|$, therefore by (i), $|1\cdot(x^{-1})|= |1|\cdot|x^{-1}|$
$1>0$, therefore $|1|=1$ and $|1|\cdot|x^{-1}|=\frac{1}{|x|}$, which is what we wish to prove.
At this point, I'm not sure you've established that $a^2=b^2$ implies that $a=\pm b$. Spivak's intent is that you work with the absolute value definition in terms of cases. So, for the first problem, break it down to the cases where $x$ and $y$ are positive, zero, or negative. It's not quite as tedious as it seems.
For the second, recall that for $a\ne 0$, $1/a$ is defined to be the unique number whose product with $a$ is $1$. So you only need to show that $|1/x|\cdot |x| = 1$; again it takes two cases.