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a. Complete just the second sentence of the proof of the lemma. Lemma. If $\lim_{n\to\infty} a(n) = a$ and $\lim_{n\to\infty} b(n) = b$, then $\lim_{n\to\infty} a(n)b(n) =$ ab. Proof. Let $\epsilon$ be a positive real number. We must show that there exists $M \in \mathbb{N}$ such for all $n \in \mathbb{N}$ with $n \ge M$, we have that (complete the sentence below) b. Consider the sequence $\frac{1}{2}, \frac{-1}{2}, \dots, \frac{(-1)^{n+1}}{2}, \dots$. Complete the already-started proof of the following lemma: Lemma. $\lim_{n\to\infty} \frac{(-1)^{n+1}}{2}$ does not exist. Proof. Suppose $L$ is a real number. We'll show $\lim_{n\to\infty} \frac{(-1)^{n+1}}{2}$ is not $L$. Let $\epsilon = \frac{1}{4}$. (Continue the proof of the lemma to completion below.)

          a. Complete just the second sentence of the proof of the lemma.
Lemma. If $\lim_{n\to\infty} a(n) = a$ and $\lim_{n\to\infty} b(n) = b$, then $\lim_{n\to\infty} a(n)b(n) =$
ab.
Proof. Let $\epsilon$ be a positive real number. We must show that there exists $M \in \mathbb{N}$
such for all $n \in \mathbb{N}$ with $n \ge M$, we have that (complete the sentence below)
b. Consider the sequence $\frac{1}{2}, \frac{-1}{2}, \dots, \frac{(-1)^{n+1}}{2}, \dots$. Complete the already-started
proof of the following lemma:
Lemma. $\lim_{n\to\infty} \frac{(-1)^{n+1}}{2}$ does not exist.
Proof. Suppose $L$ is a real number. We'll show $\lim_{n\to\infty} \frac{(-1)^{n+1}}{2}$ is not $L$.
Let $\epsilon = \frac{1}{4}$. (Continue the proof of the lemma to completion below.)

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a. Complete just the second sentence of the proof of the lemma.
Lemma. If limn→∞ a(n) = a and limn→∞ b(n) = b, then limn→∞ a(n)b(n) =
ab.
Proof. Let ϵ be a positive real number. We must show that there exists M ∈ℕ
such for all n ∈ℕ with n ≥ M, we have that (complete the sentence below)
b. Consider the sequence (1)/(2), (-1)/(2), …, ((-1)^n+1)/(2), …. Complete the already-started
proof of the following lemma:
Lemma. limn→∞((-1)^n+1)/(2) does not exist.
Proof. Suppose L is a real number. We'll show limn→∞((-1)^n+1)/(2) is not L.
Let ϵ = (1)/(4). (Continue the proof of the lemma to completion below.)

Added by William D.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Adi S

11/08/2022

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Step 1: Let's complete the sentence in the first proof: We have that |a(n)b(n) - ab| < epsilon. Show more…

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a. Complete just the second sentence of the proof of the lemma. Lemma. If lim_(n->infty )a(n)=a and lim_(n->infty )b(n)=b, then lim_(n->infty )a(n)b(n)= ab. Proof. Let epsi lon be a positive real number. We must show that there exists MinN such for all ninN with n>=M, we have that (complete the sentence below) b. Consider the sequence (1)/(2),(-1)/(2),dots,((-1)^(n+1))/(2),dots Complete the already-started proof of the following lemma: Lemma. lim_(n->infty )((-1)^(n+1))/(2) does not exist. Proof. Suppose L is a real number. We'll show lim_(n->infty )((-1)^(n+1))/(2) is not L. Let epsi lon=(1)/(4). (Continue the proof of the lemma to completion below.) a. Complete just the second sentence of the proof of the lemma Lemma. If limn--o a(n) = a and limn- b(n) = b, then limn-o a(n)b(n) = ab. Proof. Let e be a positive real number. We must show that there exists M e N such for all n e N with n > M, we have that (complete the sentence below) -1n+1 .. Complete the already-started proof of the following lemma: (-1n+1 Lemma. limn-0o does not exist. 2 (-1)n+1 Proof. Suppose L is a real number. We'll show limn- is not L 2 Let e = 1. (Continue the proof of the lemma to completion below.)

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