$\lim_{n\rightarrow \infty} (a_n+b_n)=0$ $\lim_{n\rightarrow \infty} (b_n-c_n)=0$ imply $\lim_{n\rightarrow \infty} (a_n+c_n)=0$?

Question

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?

Answer 1

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 $\lim_{n\to\infty} (a_n + c_n) = 0$.

Answer 2

No.

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