- #1
Of Mike and Men
- 54
- 3
Homework Statement
This is my second proofs course, but I've always felt uncomfortable with absolute values and inequalities because I feel like my proofs are circular or too simple. In this case, I'm not sure if I'm showing enough in my steps.
I would just like to know if this is a way to prove this. I've seen other results online, but I want to be sure I understand this.
Homework Equations
Prove if x,y are reals that |xy| = |x||y|
The Attempt at a Solution
Let x,y belong to the set of real numbers. Since |xy|=|x||y|=0 if either x or y is 0, we can assume that x,y is non-zero.
Case 1: x,y > 0
|xy| = xy = |x||y|
Case 2: x,y < 0
|xy| = (-x) (-y) = |x||y|
Case 3: Either x or y < 0, but not both
WLOG let x < 0 and y > 0 then
|xy| = (-x)(y) = |x||y|
.:. |xy| = |x||y| for x,y in the set of real numbers.