Prove if $|a_{n+1}|/|a_n| \leq |b_{n+1}|/|{b_n}|$ and $\sum |b_n|$ is convergent, then $\sum |a_n|$ is convergent How do I prove that if $|a_{n+1}|/|a_n| \leq |b_{n+1}|/|{b_n}|$ and $\sum |b_n|$ is convergent, then $\sum |a_n|$ is convergent? I know that if I can get $|b_{n+1}/{b_n}| < 1$ then I can use the ratio test to say $\sum a_n$ is convergent, but I don't know if this the proper way to approach the problem.

Hint: If $\sum b_k$ converges and successive terms in the sequence $\{b_k\}$ have greater absolute ratios, then what can you say, using the comparison test, about a sequence with smaller absolute ratios?
