Proof of the sum rule of series 
If we consider the sequence of partial sums of $ \displaystyle\sum_{n=1}^{\infty} a_n $ i.e. $S_k=\displaystyle\sum_{n=1}^{k} a_n =a_1+a_2+a_3+....+a_k$ so that $ \displaystyle\sum_{n=1}^{\infty} a_n = \lim_{k \to \infty}{S_k}=\alpha$ (as we are told that it converges).
We do the same for $b_n$ i.e. $ \displaystyle\sum_{n=1}^{\infty} b_n = \lim_{k \to \infty}{S'_k}=\beta$
Then $ \displaystyle\sum_{n=1}^{\infty}(a_n+b_n)=\lim_{k\to \infty}{(S_k+S'_k)}$ then by the regular sum rule for sequences the result follows that it converges to $ \alpha +\beta $.
Is that it or am I doing something wrong or missing some details? 
Thanks.
 A: Your proof is more or less correct, but it could use a bit more clarity and rigor. Consider the finite sums
$$\sum_{n=1}^k a_n \;\;\;\;\;\sum_{n=1}^k b_n$$
That is, let these be the partial sums of each infinite series. As $k \to \infty$ you'll have the infinite sums, presumed to converge by hypothesis. Then we get the series of equalities below (justification following):
$$\begin{align}
\sum_{n=1}^\infty (a_n + b_n) &\overset{(1)}= \lim_{k \to \infty} \sum_{n=1}^k (a_n + b_n) \\
&\overset{(2)}= \lim_{k \to \infty} \left( \sum_{n=1}^k a_n + \sum_{n=1}^k b_n \right)\\
&\overset{(3)}= \lim_{k \to \infty} \sum_{n=1}^k a_n + \lim_{k \to \infty} \sum_{n=1}^k b_n\\
&\overset{(4)}= \sum_{n=1}^\infty a_n + \sum_{n=1}^\infty b_n
\end{align}$$
Justification:

*

*$(1):$ Applying the definition of an infinite sum (limit of partial sums).

*$(2):$ Since the sum is finite, we can split it up termwise like this with no issue.

*$(3):$ As the individual limits are assumed to exist, we can split up the limit over each sum individually.

*$(4):$ Applying the definition of an infinite sum again.


Granted, I imagine you've long since surpassed need for a "proper" answer to this question, but I supposed I might as well write an answer to this unanswered question and give some help to those that need this in the future.
