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Consider some sequence $\{a_n\}_{n\in \mathbb{N}}<0$, $\{b_n\}_{n\in \mathbb{N}}>0$, $\{c_n\}_{n\in \mathbb{N}}>0$ and assume $$ \begin{cases} \lim_{n\rightarrow \infty} (a_n+b_n)=0\\ \lim_{n\rightarrow \infty} (b_n-c_n)=0 \end{cases} $$ Does this imply $$ \lim_{n\rightarrow \infty} (a_n+c_n)=0 \text{ ?} $$


I am confused because: $$ \begin{cases} (1) \hspace{1cm}\lim_{n\rightarrow \infty} (a_n+b_n)=0 \text{ is equivalent to write } a_n=-b_n+o(1)\\ (2) \hspace{1cm}\lim_{n\rightarrow \infty} (b_n-c_n)=0\text{ is equivalent to write } b_n=c_n+o(1)\\ \end{cases} $$ Put (2) in (1) and get $$ a_n=-c_n+o(1)+o(1)=-c_n+o(1) \text{ which is equivalent to write } \lim_{n\rightarrow \infty} (a_n+c_n)=0\\ $$ where $o(1)$ is a number converging to zero as $n\rightarrow \infty$

What is wrong in my arguments?

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2 Answers 2

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Yes. First, assume $\lim_{n\to\infty} (a_n + c_n) = x$ where $x \ne 0$. Now add the second equation to get $$\lim_{n\to\infty} (a_n + c_n) + \lim_{n\to\infty} (b_n - c_n) = x + 0$$ $$\lim_{n\to\infty} (a_n + b_n + c_n - c_n) = x$$ $$\lim_{n\to\infty} (a_n + b_n) = x$$ Since we already know from the equations given that $\lim_{n\to\infty} (a_n + b_n) = 0$, this leads to a contradiction proving that $x = 0$ and thus that $\lim_{n\to\infty} (a_n + c_n) = 0$.

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  • $\begingroup$ The contradiction implies either $x=0$ or $x$ does not exist. Why have you assumed $x=0$ is the true branch, doesn't that need to be proven that the limit indeed exists? $\endgroup$
    – Snared
    Jul 7, 2023 at 15:15
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No.

You could have $a_n=-n$, $b_n=c_n=n$.

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  • $\begingroup$ Thanks, could you briefly explain why my arguments above are wrong? (see added part) $\endgroup$
    – Star
    May 11, 2018 at 16:11
  • $\begingroup$ Am I missing something here? Don't these values satisfy all 3 equations $\endgroup$
    – mallan
    May 11, 2018 at 17:52
  • $\begingroup$ I do not believe this answer applies any longer. $\endgroup$
    – Kirk Fox
    May 12, 2018 at 1:01
  • $\begingroup$ This answer is wrong $\endgroup$
    – Snared
    Jul 7, 2023 at 15:14

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