To show:
Explanation of Solution
Given:
The triangle inequality
Proof:
Consider the following general triangle inequality.
Assume that the above inequality holds for any two numbers
The objective is to show that
For any
Use mathematical induction to prove the inequality.
The principle of mathematical induction states that if a certain mathematical relation is true for a base integer and, thereafter, assuming the relation to be true for an integer
Step 1: The formula holds for
Step 2: Suppose
Then,
So, by transitivity,
Hence, by principal of mathematical induction the inequality
Conclusion:
By principal of mathematical induction the inequality
Want to see more full solutions like this?
Chapter A2 Solutions
Calculus: Graphical, Numerical, Algebraic
Additional Math Textbook Solutions
Calculus & Its Applications (14th Edition)
Precalculus (10th Edition)
University Calculus: Early Transcendentals (3rd Edition)
Calculus: Early Transcendentals (3rd Edition)
Calculus and Its Applications (11th Edition)
Precalculus
- Calculus: Early TranscendentalsCalculusISBN:9781285741550Author:James StewartPublisher:Cengage LearningThomas' Calculus (14th Edition)CalculusISBN:9780134438986Author:Joel R. Hass, Christopher E. Heil, Maurice D. WeirPublisher:PEARSONCalculus: Early Transcendentals (3rd Edition)CalculusISBN:9780134763644Author:William L. Briggs, Lyle Cochran, Bernard Gillett, Eric SchulzPublisher:PEARSON
- Calculus: Early TranscendentalsCalculusISBN:9781319050740Author:Jon Rogawski, Colin Adams, Robert FranzosaPublisher:W. H. FreemanCalculus: Early Transcendental FunctionsCalculusISBN:9781337552516Author:Ron Larson, Bruce H. EdwardsPublisher:Cengage Learning