Question regarding a sequence proof from a book

  • #1
I have a Dover edition of Louis Brand's Advanced Calculus: An Introduction to Classical Analysis. I really like this book, but find his proof of limit laws for sequences questionable. He first proves the sum of null sequences is null and that the product of a bounded sequence with a null sequence is null. These proofs are fine, yet he then does this proof.

Suppose ##a_n → α## and ##b_n → β##. Write ##a_n = α + x_n## and ##b_n = β + y_n ## where ##x_n## and ##y_n## are both null sequences. Now
$$
a_n + b_n = \alpha + x_n + \beta+y_n
$$
$$
a_n + b_n -(\alpha+ \beta) = x_n + y_n
$$
We have shown ##a_n + b_n - (α+ β)## is the sum of two null sequences and therefore also null. Hence, $$a_n + b_n \to α + β.$$

Are we really allowed to rewrite a sequence to another one that converges to the same limit, i.e ##a_n = α + x_n##?
 

Answers and Replies

  • #2
34,678
6,387
I have a Dover edition of Louis Brand's Advanced Calculus: An Introduction to Classical Analysis. I really like this book, but find his proof of limit laws for sequences questionable. He first proves the sum of null sequences is null and that the product of a bounded sequence with a null sequence is null. These proofs are fine, yet he then does this proof.

Suppose ##a_n → α## and ##b_n → β##. Write ##a_n = α + x_n## and ##b_n = β + y_n ## where ##x_n## and ##y_n## are both null sequences. Now
$$
a_n + b_n = \alpha + x_n + \beta+y_n
$$
$$
a_n + b_n -(\alpha+ \beta) = x_n + y_n
$$
We have shown ##a_n + b_n - (α+ β)## is the sum of two null sequences and therefore also null. Hence, $$a_n + b_n \to α + β.$$

Are we really allowed to rewrite a sequence to another one that converges to the same limit, i.e ##a_n = α + x_n##?
It's not clear to me what you're trying to say. It's given that ##\{x_n\}## is a null sequence; i.e., ##\lim_{n \to \infty} x_n = 0##. ##a_n## doesn't have to be equal to ##\alpha + x_n##, for every n, but ##\lim_{n \to \infty} a_n = \lim_{n \to \infty} \alpha + x_n## must be true.
 
  • #3
Stephen Tashi
Science Advisor
7,578
1,467
Are we really allowed to rewrite a sequence to another one that converges to the same limit, i.e ##a_n = α + x_n##?
Of course not. We can't claim a sequence {a_n} is term-by-term equal to another abitrarily chosen sequence that converges to the same limit. However, that's not what's being done. We define ## x_n ## to be ##x_n = a_n - \alpha##. Then ##x_n## is a null sequence and, by definition, ##x_n + \alpha = a_n##.
 
  • #4
PeroK
Science Advisor
Homework Helper
Insights Author
Gold Member
2020 Award
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Are we really allowed to rewrite a sequence to another one that converges to the same limit, i.e ##a_n = α + x_n##?
You could ask the same about a number. If ##a## and ##\alpha## are numbers, then we know there is a number ##x## such that ##a = \alpha + x##.

In this case, ##a## and ##\alpha + x## represent the same number. Just as ##a_n## and ##\alpha + x_n## represent the same sequence.
 

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