Undergrad Question regarding a sequence proof from a book

[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 $a_n = α + x_n$?

 

[Mark44]

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.

 

[Stephen Tashi]

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, $x_n + \alpha = a_n$.

 

[PeroK]

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.