I'm putting together the following the proof, and I have a question about one of the final steps.
Definition of absolute value:
$\forall x \in \mathbb{R}, (x \geq 0 \Rightarrow |x| = x) \wedge (x < 0 \Rightarrow |x| = -x)$
We want to prove:
$\forall a, r, x \in \mathbb{R}, |x – a| < r \Rightarrow (a – r < x < a + r)$
Let $a, r, x, \in \mathbb{R}$. We assume $|x – a| < r$. We want to prove that $a – r < x < a +r$.
To do so, we will divide our proof into two cases.
Case 1: $x – a \geq 0$.
Then, by the definition of absolute value, $|x – a| = x – a$. So, by our assumption, $x – a < r \Longleftrightarrow x < a + r$.
Case 2: $x – a < 0$.
Then, by the definition of absolute value, $|x – a| = -x + a$. So, by our assumption, $-x + a < r \Longleftrightarrow a – r < x$.
Now, if I were to conjoin my result from case 2 with the result from case 1, I would obtain the desired statement: $(a – r < x) \wedge (x < a + r) \Longleftrightarrow a – r < x < a + r$.
My question is: how is the conjunction justified? How do I know it's okay to conjoin (use an "and" statement), rather than disjoin (use an "or" statement)?
Is it always the case that when we divide proofs into cases, the cases form a conjunction?
Thank you in advance!
Best Answer
Use the obvious fact: |z| < r implies -r < |z|.
Case 1. -r < x - a < r.
Desired conclusion follows in one step.
Case 2. -r < a - x < r.
Desired conclusion follows in two steps.