(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

If the sequence of partial sums of |a_n| is convergent and b_n is bounded, prove that the sequence of partial sums of the product (a_n)(b_n) is also convergent.

2. Relevant equations

Cauchy sequences and bounded sequences

3. The attempt at a solution

I wrote the following

for n,m > N_1 and e > 0

|a_n - a_m | < e

which proves that a_n is a cauchy sequence, for every convergent sequence is a cauchy sequence.

For b_n, we assume e = 1, and for n, m > N_2

we have

|b_n - b_m | < 1

then for N = max{N_1, N_2}

|a_n * b_n - a_m * b_m | < e + 1

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# Homework Help: Convergent Series Proof

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