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)

Let and be convergent series. For each , let and . Prove that converges.Question:

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.Solution: