I have a solution to the following problem but I am not sure whether it is correct or not. Can someone help me check my solution. Thanks! (The series in this question are all infinite series)
Question: Let and be convergent series. For each , let and . Prove that converges.
Solution: Let be the partial sums of the series respectively. Now . Similarily, . Since and converges, the sequence and converges. Since and converges to the same value, converges. Hence, the series converges.