Prove that:
a) $x \le |x|$ and
b) $|xy| = |x||y| $
Attempt:
a) If $x \ge 0: x = |x| \le |x|$.
If $ x < 0: x = |-x| = |x| \le |x|.$
b) If $xy \ge 0: |xy| = xy = |x||y|.$
If $xy < 0: |xy| = -xy = |-x||y|$ or $|x||-y| = |x||y|$.
Is there anything that is wrong?
On
If any of the variables is $0$ then the product are zero so the equality is true. Hence, we assume $x,y \ne 0$
Assume $x < 0$ and $y < 0$. Then $xy>0$ which means $$|x||y| = (-x)(-y) = xy = |xy|$$ The case $x > 0$ and $y>0$ is the same.
Assume $x>0$ and $y<0$. Then $xy < 0$ which means $$|x||y| = x(-y) = -xy = |xy|$$ The case $x<0$ and $y>0$ is the same.
Nothing is wrong. Your "attempt" is correct.
(Note that if $xy \geq 0$, we can also have $x < 0, y < 0$, but the statement still holds.)