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Chapter 7: Q. 71 (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 $a_{k} 鈫� L$ and $a_{k} 鈫� M$ , then $L = M$ .

Short Answer

Expert verified

The theorem has been proved.

Step by step solution

Step 1. Given Information

The objective is to prove that $L = M$ .

Step 2. Forming the equation.

The sequence $\{a_{k}\}$ is convergent such that $a_{k} 鈫� L$ .

By definition, for given $蔚 > 0$ ,there exists a positive integer $N$ such that

$|a_{k} - L| < 蔚$ for $k 猢� N . . . . . . . . . (1)$

The sequence $\{a_{k}\}$ is convergent such that $a_{k} 鈫� M$ .

By definition, for given $蔚 > 0$ , there exists a positive integer $P$ such that

$|a_{k} - M| < 蔚$ for $k 鈮� P . . . . . . . (2)$

Step 3. Proving the theorem

Choose $R = m a x \{N, P\}$

Therefore,

$|L - M| = |a_{k} - a_{k} + L - M| = |L - a_{k} - (M - a_{k})| 鈮� |a_{k} - L| + |a_{k} - M| < 蔚 + 蔚 = 2 蔚$

The inequality $|L - M| < 2 蔚$ holds for every $蔚 > 0$ , and $|L - M|$ is independent of $蔚$ .

Therefore,

$|L - M| = 0 L = M$

Hence, proved.

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91影视

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鈫�Land ak鈫�M, then L=M.

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Forming the equation.

Step 3. Proving the theorem

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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 $a_{k} 鈫� L$ and $a_{k} 鈫� M$ , then $L = M$ .