I am trying to prove:
Suppose that $\{a_n\}$ and $\{b_n\}$ are two sequences such that $\{a_n\}$ and $\{a_n+b_n\}$ converge. Prove that $\{b_n\}$ converges.
Here is my first attempt:
Proof: Suppose that $\{a_n\}$ and $\{b_n\}$ are two sequences such that $\{a_n\}$ and $\{a_n+b_n\}$ converge. Since $\{a_n\}$ converges, then for each $\epsilon>0$ there exist $N_1$ such that for all
$$n\geq N_1\Rightarrow|a_n-A|<\frac{\epsilon}{2}.$$
The same follows for $\{a_n+b_n\}$, so for all
$$n\ge N_2\Rightarrow|a_n+b_n-(A+B)|<\epsilon.$$
Let $N=\max\{N_1,N_2\}$. Then, we have
$$|a_n+b_n-(A+B)|=|(a_n-A)+(b_n-B)|\le|a_n-A|+|b_n-B|.$$
Since $|a_n-A|>0$, it follows that
$$|b_n-B|\lt|a_n - A|+|b_n - B|\lt\epsilon.$$
Therefore $|b_n-B|\lt\epsilon$ for $n\ge N$. Thus, $\{b_n\}$ converges.
Is it completely wrong? What can I do to improve it? Thank you.
I think the comments already made clear how to attack the problem via $\;\epsilon-\delta\;$ and etc., but arithmetic of limits makes this much simpler:
$$b_n=(a_n+b_n)-a_n$$
and thus $\;\{b_n\}\;$ converges as it is the difference of two converging limits, and its limit is the difference
$$\;\lim_{n\to\infty}(a_n+b_n) - \lim_{n\to\infty}a_n$$