Rearranging $\{p_j\}$ and $\{q_j\}$ into one sequence $\{r_j\}$ with $s_n \geqslant 0$ and $\lim_{n\to\infty} s_n = 0$ I'm trying to prove Lemma $2$ in this paper. The statement is as follows:

Let $\{p_j\}$ be a sequence (possibly finite) of non-negative numbers, and $\{-q_j\}$ be an infinite sequence of negative numbers such that $\sum p_j = \sum q_j \leqslant \infty$, $q_j \to 0$ as $j\to\infty$, and $p_j \to 0$ in case $\{p_j\}$ is an infinite sequence. Then these two sequences can be arranged into one sequence $\{r_j\}$ such that 

*

*$s_n = r_1 + \ldots + r_n \geqslant 0$ for all $n$, and

*$s_n\to 0$ as $n\to\infty$.


If any $p_i$ equals $0$, we can simply discard it since it doesn't alter any of the sums. Now, put $r_1 := p_1 > 0$. If $q_1 < p_1$, put $r_2 := -q_1$. Otherwise, keep looking until you find a $q_k$ such that $q_k < p_1$ and put $r_2 := -q_k$. We can find such a $q_k$ since $\lim_{j\to\infty} q_j = 0$. I could keep picking terms from either sequence and (hopefully) ensure that $s_n \geqslant 0$ while $s_n \to 0$, but it is not clear why this process would work and terminate.
My idea for a proof is constructive/algorithmic, but I couldn't see it to completion. Perhaps there is a better, non-constructive approach? Thanks a lot!
 A: Before starting the answer, I would like to say that, $p_k$ is finite is equivalent to say that for some integer $n$, $p_k = 0$ whenever $k\ge n$.
I would start by using induction to construct a strictly increasing sequence of intergers $\left(m_\ell\right)_{\ell\in \mathbb N}$ such that $m_0 = 0$

$$\sum_{k=1}^{m_\ell} p_k -\sum_{j=1}^\ell q_j > 0;\; \forall \ell\in \mathbb N_{\ge 1}$$

First, since $$\sum_{k=1}^\infty p_k - q_1 = \sum_{j=2}^\infty q_j > 0$$ then one cann obtain an integer $m_1$ such that, $$\sum_{k=1}^{m_1} p_k - q_1 > 0.$$
Now assume that you have constructed $m_1, m_2, \ldots, m_\ell$. To construct $m_{\ell + 1}$ use the fact that:
$$\sum_{k=1}^\infty p_k - \sum_{j=1}^\ell q_j = \sum_{j=\ell+1}^\infty q_j > 0.$$
This ended the induction.
Now let $$r_k =\begin{cases}p_{k-\ell} & \text{if $m_\ell + \ell < k \le m_{\ell + 1} + \ell$}\\
-q_{\ell + 1} & \text{if $k = m_{\ell} + \ell$}
\end{cases}$$
Let $n\in \mathbb N_{\ge 1}$ and $\ell$ such that $m_\ell + \ell < n \le m_{\ell+1} +\ell$
\begin{align}
\sum_{k=1}^n r_k &\ge \sum_{k=1}^{m_\ell} p_k - \sum_{j=1}^\ell q_j \ge 0
\end{align}
this proves $s_n \ge 0$. For the limit:
\begin{align}
\sum_{k=1}^\infty r_k &= \lim\limits_{\ell \to\infty}\left(\sum_{k=1}^{m_\ell} p_k - \sum_{j=1}^\ell q_j\right) = 0.
\end{align}

Basically the idea is just at each time add a bunch of terms $p_j$ that will ensure when adding the next term $-q_j$ you will have a non-negative value.
