How to show the additive property of a sum of a series

Question

So I have been working on this proof for a bit now. I found a way to prove it directly, but am struggling to find a way to prove it with limits.

Question:

Let ∑an = A, ∑bn = B, for an, bn, A, B in the Reals,

show that ∑(an+bn) = (A+B)

Answer 1

If you have two convergent sequences, the limit of the sum is the sum of the limits. So you have \begin{align} \sum_{n=1}^\infty(a_n+b_n)&=\lim_{m\to\infty}\sum_{n=1}^m(a_n+b_n)=\lim_{m\to\infty}\left(\sum_{n=1}^ma_n+\sum_{n=1}^mb_n\right)\\ \ \\ &=\lim_{m\to\infty}\sum_{n=1}^ma_n+\lim_{m\to\infty}\sum_{n=1}^mb_n =\sum_{n=1}^\infty a_n+=\sum_{n=1}^\infty b_n \end{align}