Chapter 7: Q. 74 (page 605)

Let $\{a_{k}\}$ be a sequence. Prove the indicated limit rules from Theorem 7.12. You may wish to model your proofs on the proofs of the analogous statements from Section 1.5.
Prove that if $|a_{k}| \to 0$ , then $a_{k} \to 0$ .

Short Answer

Expert verified

The theorem is thus proved.

Step by step solution

Step 1. Given Information.

The objective is to prove that $a_{k} \to 0$ .

Step 2. Assumption

The sequence $\{a_{k}\}$ is convergent and converges to $0$ .

By definition of convergence, for $ε > 0$ there is a positive integer $N$ , such that

$||a_{k}| - 0| < ε f o r k \geq N ||a_{k}|| < ε |a_{k}| < ε (B e c a u s e |a_{k}| \geq 0)$

Step 3. Proving the theorem.

For $ε > 0$ , there is a positive integer $N$ such that,

$|a_{k}| < ε f o r k \geq N |a_{k} - 0| < ε f o r k \geq N$

Thus, a positive integer $N$ can be found such that $|a_{k} - 0| < ε$ .

Therefore, $\lim_{k \to \infty} a_{k} = 0$

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Let $\{a_{k}\}$ be a sequence. Prove the indicated limit rules from Theorem 7.12. You may wish to model your proofs on the proofs of the analogous statements from Section 1.5.
Prove that if $|a_{k}| \to 0$ , then $a_{k} \to 0$ .

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Assumption

Step 3. Proving the theorem.

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Let akbe a sequence. Prove the indicated limit rules from Theorem 7.12. You may wish to model your proofs on the proofs of the analogous statements from Section 1.5.Prove that if ak→0, then ak→0.

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Assumption

Step 3. Proving the theorem.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

Decision Maths

Discrete Mathematics

Statistics

Geometry

Applied Mathematics

Study anywhere. Anytime. Across all devices.

Let $\{a_{k}\}$ be a sequence. Prove the indicated limit rules from Theorem 7.12. You may wish to model your proofs on the proofs of the analogous statements from Section 1.5.
Prove that if $|a_{k}| \to 0$ , then $a_{k} \to 0$ .