Let $(a_n)_{n\in\mathbb{N}}, (e_n)_{n\in\mathbb{N}}, (p_n)_{n\in\mathbb{N}}, (r_n)_{n\in\mathbb{N}}$ be nonnegative sequences in $\mathbb{R}$ with $(a_n)_{n\in\mathbb{N}}\in\ell^1$, $(e_n)_{n\in\mathbb{N}}\in\ell^1$, and $(p_n)_{n\in\mathbb{N}}$ a convergent sequence. Furthermore, assume that $(1-a_n)p_{n+1}-p_n+r_n\leq e_n$ holds for all $n$. I want to show that $(r_n)_{n\in\mathbb{N}}\in\ell^1$. Here is my attempt at a proof:
$\sum\limits_{n\in\mathbb{N}} |r_n| = \sum\limits_{n\in\mathbb{N}} r_n \leq \sum\limits_{n\in\mathbb{N}}e_n+p_n+(a_n-1)p_{n+1}=\sum\limits_{n\in\mathbb{N}}e_n +\sum\limits_{n\in\mathbb{N}}(p_n-p_{n+1}) +\sum\limits_{n\in\mathbb{N}}a_np_n$.
Since $(p_n)_{n\in\mathbb{N}}$ converges, it is bounded above by some $M\in \mathbb{R}$. Using this we can write,
$\sum\limits_{n\in\mathbb{N}}e_n +p_1-\lim\limits_{n\rightarrow\infty}p_n +\sum\limits_{n\in\mathbb{N}}a_np_n\leq \sum\limits_{n\in\mathbb{N}}e_n +p_1-\lim\limits_{n\rightarrow\infty}p_n +M\sum\limits_{n\in\mathbb{N}}a_n<\infty$.
The sum $\sum\limits_{n\in\mathbb{N}}e_n$ is finite as $(e_n)_{n\in\mathbb{N}}$ is nonnegative in $\ell^1$ and similarly for the sum $\sum\limits_{n\in\mathbb{N}}a_n$. Thus $(r_n)_{n\in\mathbb{N}}\in\ell^1$.
Is this sufficient?