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.