And incase it interests anyone, the two prior theorems motivate a proof of the triangle inequality, which I also figure I'll include :
Theorem (triangle inequality)- For all real numbers x and y, |x + y| ≤ |x| + |y|.
Proof- Let x and y be arbitrary real numbers. Note that x + y is also a real number, as is |x| + |y|. Using these notions, along with our first theorem above (for all real numbers a and b, |a|≤ b iff -b ≤ a ≤ b), letting a = x + y and b = |x| + |y|), we have |x + y| ≤ (|x| + |y|) iff -(|x| + |y|) ≤ x + y ≤ (|x| + |y|). Now, by our second theorem above (for any real number c, -|c|≤ c ≤ |c|, applying it once for c = x and again for c = y), we have both -|x|≤ x ≤ |x| and -|y|≤ y ≤ |y|. This means -|x|≤ x, -|y|≤ y, x ≤ |x|, and y ≤ |y|. Adding -|x|≤ x to -|y|≤ y yields -|x|-|y| ≤ x + y. Adding x ≤ |x|, to y ≤ |y| yields x + y ≤ |x|+|y|. If we then, in turn, add these resulting inequalities, we are left with -|x|-|y| ≤ x + y ≤ |x|+|y|, or otherwise, -(|x|+|y|) ≤ x + y ≤ |x|+|y|, as desired.