Sequences of non-negative terms.

Question

Prove that if $\{a_{n}\}$ and $\{b_{n}\}$ are sequences of non-negative terms, $\sum a_{n}$ converges and $\lim_{n\to\infty}b_{n}=0$ then $\sum a_{n}b_{n}$ converges.

I don't understand how to prove it, can you help me? please.

Answer 1

We have: $b_n \le 1, n \ge K \implies a_nb_n \le a_n, n \ge K\implies \sum_{n} a_nb_n \le \sum_{n} a_n\implies \sum_{n} a_nb_n$ converges by comparison test.