Question

Exercise 4.1. Prove that the subadditivity property of the absolute value is true. That is, for all a, b eR, \a] + [6] > |a +b].

Answer 1

To proof:

Subadditivity property, we use definition of absolute value of a number.

The absolute value of a number is      x if x > 0 (*),

                                                          –x if x < 0 (**)

                                                            0 if x = 0 (**)

Case 1: a and b are positive If a and b are both positive, then a + b and |a + b| are also positive. From *,

|a| + |b| = a + b = |a + b|.

Case 2 : a and b are both negative If a and b are both negative, from **, |a| + |b| = (-a) + (-b) = -(a + b) But from **, |x| = –x if x <0, so -(a + b) = |a + b| which means that

|a| + |b| = |a + b|.

Case 3: a is positive and b is negative |a| + |b| = a + (-b) and |a + b| is a + b itself (if positive) or –(a + b) if negative. But, a > –a and – b > b so,

a + (-b) > a + b and (-a) + (-b) > -(a + b).

In either case, we have |a| + |b| > |a + b|. Therefore, combining the three cases, we have

|a| + |b| ≥ |a + b|