$\displaystyle \{a_n\}$ and $\displaystyle \{a_n+b_n\}$ converge. Prove $\displaystyle \{b_n\}$ converges.

Let $\displaystyle \{a_n\}$ and $\displaystyle \{a_n+b_n\}$ converge to $\displaystyle A, \ A+\alpha$, respectively,

Then there is an $\displaystyle \epsilon>0$ and $\displaystyle \exists N\in\mathbb{N}, \ n\geq N$,

$\displaystyle |a_n-A|<\frac{\epsilon}{2}$

and

$\displaystyle |a_n+b_n-(A+\alpha)|\leq \epsilon$

$\displaystyle |a_n+b_n-(A+\alpha)|\leq |a_n-A|+|b_n-\alpha|<\frac{\epsilon}{2}+|b_n-\alpha|<\epsilon$

$\displaystyle \Rightarrow |b_n-\alpha|<\frac{\epsilon}{2}<\epsilon$

Therefore, $\displaystyle \{b_n\}$ converges.

Correct?