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: show that $|a_n|\le|b_n|\dfrac{|a_1|}{|b_1|}$.
