# Question regarding a sequence proof from a book

• I
• MathMorlock
The only difference is that ##a_n## is a sequence that converges to ##\alpha## while ##\alpha + x_n## is a sequence that converges to ##x##.f

#### MathMorlock

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##?

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.

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##.

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.