$(a_n)_{n=1}^\infty$ & $(b_n)_{n=1}^\infty$ are seq st $(a_n)_{n=1}^\infty$ & $[{(a_n)_{n=1}^\infty + (b_n)_{n=1}^\infty}]$ con. Prove $(b_n)$ con Suppose $(a_n)_{n=1}^\infty$ and $(b_n)_{n=1}^\infty$ are sequences such that $(a_n)_{n=1}^\infty$ and $[{(a_n)_{n=1}^\infty + (b_n)_{n=1}^\infty}]$ converge. Prove that $(b_n)_{n=1}^\infty$ converges
I can say $b_n=(a_n + b_n)-a_n$. Since both $(a_n)_{n=1}^\infty$ and $[{(a_n)_{n=1}^\infty + (b_n)_{n=1}^\infty}]$ converge, isnt' there a subtraction rule that says that because those both converge, that $b_n$ would also converge?

hint
Given $ \epsilon>0 $ ,Assume
$$\lim_{n\to+\infty}a_n=A$$
and
$$\lim_{n\to+\infty}(a_n+b_n)=C$$
then
$$|b_n-(C-A)|=$$
$$|(a_n+b_n)-C-(a_n-A)|$$
$$\le |(a_n+b_n)-C|+|a_n-A|$$
$$<\frac{\epsilon}{2}+\frac{\epsilon}{2}$$
for $ n $ large enough.
