Prove |a + b + c| ≤ |a| + |b| + |c| for all a, b, c ∈ R. Hint : Apply the
triangle inequality twice. Do not consider eight cases.
3.1 (a) Which of the properties A1–A4, M1–M4, DL, O1–O5 fail for N?
(b) Which of these properties fail for Z?
3.2 (a) The commutative law A2 was used in the proof of (ii) in
Theorem 3.1. Where?
(b) The commutative law A2 was also used in the proof of (iii) in
Theorem 3.1. Where?
3.3 Prove (iv) and (v) of Theorem 3.1.
3.4 Prove (v) and (vii) of Theorem 3.2.
3.5 (a) Show |b| ≤ a if and only if −a ≤ b ≤ a.
(b) Prove ||a| − |b|| ≤ |a − b| for all a, b ∈ R.
3.6 (a) Prove |a + b + c| ≤ |a| + |b| + |c| for all a, b, c ∈ R. Hint : Apply the
triangle inequality twice. Do not consider eight cases.
(b) Use induction to prove
|a1 + a2 + · · · + an| ≤ |a1| + |a2| + · · · + |an|
for n numbers a1, a2, . . . , an.
3.7 (a) Show |b| < a if and only if −a < b < a.
(b) Show |a − b| < c if and only if b − c < a < b + c.
(c) Show |a − b| ≤ c if and only if b − c ≤ a ≤ b + c.
3.8 Let a, b ∈ R. Show if a ≤ b1 for every b1 > b, then a ≤ b.
