Let (an) and (bn) be convergent sequences such that lim an = L₁ and lim b,= L₂. Prove, using 71-00 11400 N. e type methods only, that the limit operator is linear and, distributes across products such that lim (a, b) L₁ L₂ 11-00
Let (an) and (bn) be convergent sequences such that lim an = L₁ and lim b,= L₂. Prove, using 71-00 11400 N. e type methods only, that the limit operator is linear and, distributes across products such that lim (a, b) L₁ L₂ 11-00
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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![### Problem Statement
Let \(\{a_n\}\) and \(\{b_n\}\) be convergent sequences such that
\[ \lim_{{n \to \infty}} a_n = L_1 \]
and
\[ \lim_{{n \to \infty}} b_n = L_2. \]
Prove, **using \(N, \epsilon \) type methods only**, that the limit operator is linear and distributes across products, such that
\[ \lim_{{n \to \infty}} (a_n \cdot b_n) = L_1 \cdot L_2. \]
### Key Points
- **Convergent Sequences:** Sequences \(\{a_n\}\) and \(\{b_n\}\) converge to \(L_1\) and \(L_2\) respectively.
- **Limit Operator Linearity:** The objective is to show that the limit operator distributes across the product of two convergent sequences using \(N, \epsilon \) methods.
### Methodology
1. **Definitions:**
- A sequence \(\{a_n\}\) converges to \(L_1\) means for every \(\epsilon > 0\), there exists a positive integer \(N_1\) such that for all \(n \geq N_1\), \(|a_n - L_1| < \epsilon\).
- Similarly, \(\{b_n\}\) converges to \(L_2\) means for every \(\epsilon > 0\), there exists a positive integer \(N_2\) such that for all \(n \geq N_2\), \(|b_n - L_2| < \epsilon\).
2. **Using \( N, \epsilon \) Methods:**
- Combine the definitions of limits for each sequence in order to derive the relationship involving the products \(a_n \cdot b_n\).
- Use properties of absolute values and algebraic manipulation to achieve the desired result.
3. **Proof Structure:**
- Identify suitable \(N\) and \(\epsilon\) for proving the product convergence.
- Formulate the proof to show \(\lim_{{n \to \infty}} (a_n \cdot b_n)\).
### Goal
Demonstrate that according to the defined properties](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcdbdf4c8-5459-45de-99f8-cd8d71031beb%2Fa6f73b55-ad5a-4f55-b809-6d1a1b955de7%2Flfwe4ry_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Problem Statement
Let \(\{a_n\}\) and \(\{b_n\}\) be convergent sequences such that
\[ \lim_{{n \to \infty}} a_n = L_1 \]
and
\[ \lim_{{n \to \infty}} b_n = L_2. \]
Prove, **using \(N, \epsilon \) type methods only**, that the limit operator is linear and distributes across products, such that
\[ \lim_{{n \to \infty}} (a_n \cdot b_n) = L_1 \cdot L_2. \]
### Key Points
- **Convergent Sequences:** Sequences \(\{a_n\}\) and \(\{b_n\}\) converge to \(L_1\) and \(L_2\) respectively.
- **Limit Operator Linearity:** The objective is to show that the limit operator distributes across the product of two convergent sequences using \(N, \epsilon \) methods.
### Methodology
1. **Definitions:**
- A sequence \(\{a_n\}\) converges to \(L_1\) means for every \(\epsilon > 0\), there exists a positive integer \(N_1\) such that for all \(n \geq N_1\), \(|a_n - L_1| < \epsilon\).
- Similarly, \(\{b_n\}\) converges to \(L_2\) means for every \(\epsilon > 0\), there exists a positive integer \(N_2\) such that for all \(n \geq N_2\), \(|b_n - L_2| < \epsilon\).
2. **Using \( N, \epsilon \) Methods:**
- Combine the definitions of limits for each sequence in order to derive the relationship involving the products \(a_n \cdot b_n\).
- Use properties of absolute values and algebraic manipulation to achieve the desired result.
3. **Proof Structure:**
- Identify suitable \(N\) and \(\epsilon\) for proving the product convergence.
- Formulate the proof to show \(\lim_{{n \to \infty}} (a_n \cdot b_n)\).
### Goal
Demonstrate that according to the defined properties
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