Let {$a_n$} and {$p_n$} be two sequences with $p_n>0$ for all $n\in \mathbb{N}$.
(i) Suppose that for all $n\in \mathbb{N}$, $a_n\neq 0$, and $\left|\frac{a_{n+1}}{a_n}\right|\leq \frac{p_{n+1}}{p_n}$. Prove that if $\sum_{n}p_n$ is convergent, then $\sum_{n}|a_n|$ is convergent.
Initially, I thought Theorem 3.25 (a) in Baby Rudin, i.e.
If $|a_n|\leq c_n$ for $n\geq N_0$, where $N_0$ is some fixed integer, and if $\sum_{n}c_n$ converges, then $\sum_{n}a_n$ converges.
would work. However, I noticed it is difficult to use this to prove the absolute convergence like this problem. What approach or theorem does work?
Similarly,
(ii) Suppose that $\lim\limits_{n\to\infty} \frac{|a_n|}{p_n}=l\in (0,\infty)$. Prove that the series $\sum_{n}|a_n|$ and $\sum_{n}p_n$ are either both divergent or both convergent.
I have been looking for a theorem that can directly apply to this (ii) in the same text, however, I have no idea.