To prove that the sum of two convergent series $a_k$ and $b_k$ converges, we will use the fact that the sum of two Cauchy sequences is also a Cauchy sequence.

Step-by-Step Proof:

1. Recall the definition of convergence for series:

   A series $\sum a_k$ converges if the sequence of partial sums $S_N = \sum_{k=1}^N a_k$ converges to a limit as $N \to \infty$. Similarly, for the series $\sum b_k$, the partial sums $T_N = \sum_{k=1}^N b_k$ converge to a limit.

2. Convergence implies Cauchy sequence:

   If a series $\sum a_k$ converges, then the sequence of its partial sums $S_N$ is a Cauchy sequence. That is, for any $\epsilon > 0$, there exists an integer $N_1$ such that for all $n, m > N_1$, $|S_n - S_m| < \epsilon.$ Similarly, since $\sum b_k$ converges, the sequence of partial sums $T_N$ is also a Cauchy sequence. That is, for any $\epsilon > 0$, there exists an integer $N_2$ such that for all $n, m > N_2$, $|T_n - T_m| < \epsilon.$

3. Sum of two convergent series:

   Consider the series $\sum (a_k + b_k)$, where we define the partial sums of the sum as $U_N = \sum_{k=1}^N (a_k + b_k)$. We can write this as: $U_N = S_N + T_N,$ where $S_N$ is the partial sum of $\sum a_k$, and $T_N$ is the partial sum of $\sum b_k$.

4. Show that $U_N$ is a Cauchy sequence:

   To show that $\sum (a_k + b_k)$ converges, we need to prove that the sequence $U_N$ is Cauchy. Let $n, m > \max(N_1, N_2)$. Then: $|U_n - U_m| = |(S_n + T_n) - (S_m + T_m)| = |(S_n - S_m) + (T_n - T_m)|.$ Using the triangle inequality: $|U_n - U_m| \leq |S_n - S_m| + |T_n - T_m|.$ Since $|S_n - S_m| < \epsilon$ and $|T_n - T_m| < \epsilon$ for $n, m > N_1, N_2$, we have: $|U_n - U_m| < \epsilon + \epsilon = 2\epsilon.$ This shows that $U_N$ is a Cauchy sequence.

5. Conclusion:

   Since the sequence of partial sums $U_N$ is Cauchy, and since every Cauchy sequence in $\mathbb{R}$ converges, it follows that the series $\sum (a_k + b_k)$ converges.

Thus, the sum of two convergent series $\sum a_k$ and $\sum b_k$ converges.

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Would you like to go over the steps in more detail? Let me know if you have any questions.

Related Questions:

1. What does it mean for a series to converge absolutely?
2. How can you prove that a sequence is Cauchy?
3. What are the implications of the Cauchy Criterion for series?
4. How do we prove that the difference of two convergent series also converges?
5. What is the relationship between Cauchy sequences and limits?

Tip: When working with series, remember that convergence implies the existence of a limit, and that partial sums play a crucial role in proving convergence.