Introduction to Mathematical Analysis I - Second Edition

21 Figure 1.1: The absolute value function. The following properties of absolute value follow directly from the definition. Proposition 1.4.2 Let x , y , M ∈ R and suppose M > 0. The following properties hold: (a) | x | ≥ 0; (b) | − x | = | x | ; (c) | xy | = | x || y | ; (d) | x | < M if and only if − M < x < M . (The same holds if < is replaced with ≤ .) Proof: We prove (d) and leave the other parts as an exercise. (d) Suppose | x | < M . In particular, this implies M > 0. We consider the two cases separately: x ≥ 0 and x < 0. Suppose first x ≥ 0. Then | x | = x and, hence, − M < 0 ≤ x = | x | < M . Now suppose x < 0. Then | x | = − x . Therefore, − x < M and, so x > − M . It follows that − M < x < 0 < M . For the converse, suppose − M < x < M . Again, we consider different cases. If x ≥ 0, then | x | = x < M as desired. Next suppose x < 0. Now, − M < x implies M > − x . Then | x | = − x < M . Note that as a consequence of part (d) above, since | x | ≤ | x | we get −| x | ≤ x ≤ | x | . The next theorem will play an important role in the study of limits. Theorem 1.4.3 — Triangle Inequality. Given x , y ∈ R , | x + y | ≤ | x | + | y | . Proof: From the observation above, we have −| x | ≤ x ≤ | x | −| y | ≤ y ≤ | y | . Adding up the inequalities gives −| x | − | y | ≤ x + y ≤ | x | + | y | .