The following proposition is clearly wrong. Provide a counterexample, and explain what is wrong with the proof. Proposition. For all n ≥ 1, for all a₁,..., an € R, we have a₁ = A2 = ... = an. Proof. We will prove by induction on n. Base Case: For n = 1, let a₁ € R. Then a₁ = a₁. 1 Inductive Step: Assume that for all a₁,... an ER, we have a₁ = = an. We will prove that for all a₁,..., an+1 € R, we have an+1. a2 = a₁ Let a₁,..., an+1 be in R. Observe that • Since a₁,... an are n real numbers, by inductive hypothesis, we know that a₁. = An. • Since a2,...an+1 are n real numbers, by inductive hypothesis, we know that a2 = ··· = An = An+1. (Note that the inductive hypothesis is a for-all statement, so we can apply it to any n real numbers.
The following proposition is clearly wrong. Provide a counterexample, and explain what is wrong with the proof. Proposition. For all n ≥ 1, for all a₁,..., an € R, we have a₁ = A2 = ... = an. Proof. We will prove by induction on n. Base Case: For n = 1, let a₁ € R. Then a₁ = a₁. 1 Inductive Step: Assume that for all a₁,... an ER, we have a₁ = = an. We will prove that for all a₁,..., an+1 € R, we have an+1. a2 = a₁ Let a₁,..., an+1 be in R. Observe that • Since a₁,... an are n real numbers, by inductive hypothesis, we know that a₁. = An. • Since a2,...an+1 are n real numbers, by inductive hypothesis, we know that a2 = ··· = An = An+1. (Note that the inductive hypothesis is a for-all statement, so we can apply it to any n real numbers.
College Algebra (MindTap Course List)
12th Edition
ISBN:9781305652231
Author:R. David Gustafson, Jeff Hughes
Publisher:R. David Gustafson, Jeff Hughes
Chapter8: Sequences, Series, And Probability
Section8.5: Mathematical Induction
Problem 42E
Related questions
Question
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 3 steps with 3 images
Recommended textbooks for you
College Algebra (MindTap Course List)
Algebra
ISBN:
9781305652231
Author:
R. David Gustafson, Jeff Hughes
Publisher:
Cengage Learning
College Algebra
Algebra
ISBN:
9781305115545
Author:
James Stewart, Lothar Redlin, Saleem Watson
Publisher:
Cengage Learning
College Algebra (MindTap Course List)
Algebra
ISBN:
9781305652231
Author:
R. David Gustafson, Jeff Hughes
Publisher:
Cengage Learning
College Algebra
Algebra
ISBN:
9781305115545
Author:
James Stewart, Lothar Redlin, Saleem Watson
Publisher:
Cengage Learning
Elements Of Modern Algebra
Algebra
ISBN:
9781285463230
Author:
Gilbert, Linda, Jimmie
Publisher:
Cengage Learning,
Algebra and Trigonometry (MindTap Course List)
Algebra
ISBN:
9781305071742
Author:
James Stewart, Lothar Redlin, Saleem Watson
Publisher:
Cengage Learning