Chapter 7: Q. 88 (page 605)

Prove Theorem 7.14. That is, show that if $\{a_{k}\}$ is a sequence that converges to L, then every subsequence of $\{a_{k}\}$ also converges to L

Short Answer

Expert verified

Proved that every subsequence of the sequence $\{a_{k}\}$ converges to the same limit L

Step by step solution

Step 1. Given information

The given sequence $\{a_{k}\}$ converging to limit L

Step 2. Finding the subsequence of ak

For given $ε > 0$ ,there exists a positive integer Nsuch that

$|a_{k} - L| < ε f o r k \geq N$

Since $\{a_{k}\}$ is a subsequence of sequence $\{a_{k}\}$ ; therefore

$n_{k} \geq k$

The inequalities $k \geq N a n d n_{k} \geq k i m p l i e s n_{k} \geq k$

Thus,for given $ε > 0$ ,there exists a positive integer N such that

$|a_{n_{k}} - L| < ε f o r n_{k} \geq k$

Thus,subsequence $\{a_{n_{k}}\}$ converges to L

Therefore,every subsequence of the sequence $\{a_{k}\}$ converges to the same limitL

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Prove Theorem 7.14. That is, show that if $\{a_{k}\}$ is a sequence that converges to L, then every subsequence of $\{a_{k}\}$ also converges to L

Short Answer

Step by step solution

Step 1. Given information

Step 2. Finding the subsequence of ak

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Discrete Mathematics

Theoretical and Mathematical Physics

Probability and Statistics

Calculus

Logic and Functions

Applied Mathematics

Study anywhere. Anytime. Across all devices.

Prove Theorem 7.14. That is, show that if ak is a sequence that converges to L, then every subsequence ofak also converges to L

Short Answer

Step by step solution

Step 1. Given information

Step 2. Finding the subsequence of ak

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Discrete Mathematics

Theoretical and Mathematical Physics

Probability and Statistics

Calculus

Logic and Functions

Applied Mathematics

Study anywhere. Anytime. Across all devices.

Prove Theorem 7.14. That is, show that if $\{a_{k}\}$ is a sequence that converges to L, then every subsequence of $\{a_{k}\}$ also converges to L