Basic real analysis - Proving convergence of series

Question

I am not sure if I'm heading in the right direction for this proof. I started off with the fact that a series is convergent if it satisfies the Cauchy criterion so I have to show that the sequence a_n satisfies the Cauchy criterion. I begin to prove, "If the series b_n converges then the series a_n converges. Since b_n converges then b_n satisfies the Cauchy criterion and there exists a b_m such that the sequence b_n converges to. From here I don't know where to go to show a_n satisfies the Cauchy criterion.

Cauchy criterion:

Answer 1

Suppose $\lim(a_n/b_n)=C\neq0.$

($\Rightarrow$)Assume $\sum a_n$ converge,

let there exists $K\in \Bbb N$ such that $C/2\le a_n/b_n \le 2C$, $\forall n\ge K$, i.e. $$(C/2)b_n\le a_n\le 2Cb_n\text{ , } \forall n\ge K$$ $$(C/2)\sum_{n=K}^{\infty}b_n \le \sum_{n=K}^{\infty}a_n\le 2C\sum_{n=K}^{\infty}b_n$$ $\sum a_n=\sum_{n=1}^{K-1}a_n+\sum_{n=K}^{\infty}a_n$. So $\sum_{n=K}^{\infty}b_n$ converge, hence $\sum b_n$ converge.

Similar argument for another direction.