Basic real analysis - Proving convergence of series 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 an satisfies the Cauchy criterion. I begin to prove, "If the series bn converges then the series an converges. Since bn converges then bn satisfies the Cauchy criterion and there exists a bm such that the sequence bn converges to. From here I don't know where to go to show an satisfies the Cauchy criterion.

Cauchy criterion:
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.
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.