4. Prove that -(lal+ |6|) < a +b< |a| + [b| for any real numbers a and b.(Hint: Use cases when a, b > 0; a, b < 0; and a & b have different signs.)5.Prove the triangle inequality given by la + b| < la| + 16|.6. Use mathematical induction to show that for any positive integer n,(1+2+ .+n)² = 1³ + 2° + ... + n°.(Hint: Apply binomial expansion)

Answered: 4. Prove that (a + b) ≤a+b≤ a + b for…

4. Prove that (a + b) ≤a+b≤ a + b for any real numbers a and b. (Hint: Use cases when a, b ≥ 0; a, b < 0; and a & b have different signs.)

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter2: The Integers

Section2.2: Mathematical Induction

Problem 49E: Show that if the statement is assumed to be true for , then it can be proved to be true for . Is...

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Question

4. Prove that -(lal+ |6|) < a +b< |a| + [b| for any real numbers a and b.
(Hint: Use cases when a, b > 0; a, b < 0; and a & b have different signs.)
5.
Prove the triangle inequality given by la + b| < la| + 16|.
6. Use mathematical induction to show that for any positive integer n,
(1+2+ .+n)² = 1³ + 2° + ... + n°.
(Hint: Apply binomial expansion)

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