Question about series rearrangement in Baby Rudin (theorem 3.54). I have trouble following the following theorem in Rudin:

Let $\sum a_n$ be a series of real numbers which converges, but not absolutely. Suppose
$$-\infty \leq \alpha \leq \beta \leq +\infty.$$
Then there exists a rearrangement $\sum a_n^\prime$ with partial sums $s_n^\prime$ such that
$$\lim_{n\to\infty}\inf s_n^\prime = \alpha, \ \ \ \mbox{ and } \ \ \ \lim_{n\to\infty}\sup s_n^\prime = \beta. \tag{24}$$

Here's the proof:

Let $$p_n = \frac{|a_n| + a_n}{2}, \ q_n = \frac{|a_n| - a_n}{2} \ (n = 1, 2, 3, \ldots). $$
Then $p_n - q_n = a_n$, $p_n + q_n = |a_n|$, $p_n \geq 0$, $q_n \geq 0$. The series $\sum p_n$, $\sum q_n$ must both diverge.


For if both were convergent, then $$\sum \left( p_n + q_n \right) = \sum |a_n|$$ would converge, contrary to hypothesis. Since $$ \sum_{n=1}^N a_n = \sum_{n=1}^N \left( p_n - q_n \right) = \sum_{n=1}^N p_n - \sum_{n=1}^N q_n,$$ divergence of $\sum p_n$ and convergence of $\sum q_n$ (or vice versa) implies divergence of $\sum a_n$, again contrary to hypothesis.


Now let $P_1, P_2, P_3, \ldots$ denote the non-negative terms of $\sum a_n$, in the order in which they occur, and let $Q_1, Q_2, Q_3, \ldots$ be the absolute values of the negative terms of $\sum a_n$, also in their original order.


The series $\sum P_n$, $\sum Q_n$ differ from $\sum p_n$, $\sum q_n$ only by zero terms, and are therefore divergent.


We shall construct sequences $\{m_n \}$, $\{k_n\}$, such that the series $$ P_1 + \cdots + P_{m_1} - Q_1 - \cdots - Q_{k_1} + P_{m_1 + 1} + \cdots + P_{m_2} - Q_{k_1 + 1} - \cdots - Q_{k_2} + \cdots \tag{25}, $$ which clearly is a rearrangement of $\sum a_n$, satisfies (24).


Choose real-valued sequences $\{ \alpha_n \}$, $\{ \beta_n \}$ such that $\alpha_n \rightarrow \alpha$, $\beta_n \rightarrow \beta$, $\alpha_n < \beta_n$, $\beta_1 > 0$.


Let $m_1$, $k_1$ be the smallest integers such that $$P_1 + \cdots + P_{m_1} > \beta_1,$$ $$P_1 + \cdots + P_{m_1} - Q_1 - \cdots - Q_{k_1} < \alpha_1;$$  let $m_2$, $k_2$ be the smallest integers such that $$P_1 + \cdots + P_{m_1} - Q_1 - \cdots - Q_{k_1} + P_{m_1 + 1} + \cdots + P_{m_2} > \beta_2,$$ $$P_1 + \cdots + P_{m_1} - Q_1 - \cdots - Q_{k_1} + P_{m_1 + 1} + \cdots + P_{m_2} - Q_{k_1 + 1} - \cdots - Q_{k_2} < \alpha_2;$$ and continue in this way. This is possible since $\sum P_n$, $\sum Q_n$ diverge.


If $x_n$, $y_n$ denote the partial sums of (25) whose last terms are $P_{m_n}$, $-Q_{k_n}$, then $$ | x_n - \beta_n | \leq P_{m_n}, \ \ \ |y_n - \alpha_n | \leq Q_{k_n}. $$ Since $P_n \rightarrow 0$, $Q_n \rightarrow 0$ as $n \rightarrow \infty$, we see that $x_n \rightarrow \beta$, $y_n \rightarrow \alpha$.


Finally, it is clear that no number less than $\alpha$ or greater than $\beta$ can be a subsequential limit of the partial sums of (25).

I don't understand the last two lines of the proof (put in bold). I'm aware that this question was already asked on this forum, but I didn't understand the answers provided in this question, so that's why I'm writing my own question.
Thanks in advance.

I will consider the case $\beta \in \mathbb{R}$ here.
Rudin showed that $x_n \to \beta$ on p.77.
Let $\epsilon$ be an arbitrary positive real number.
Then, there exists a natural number $N$ such that $$n \geq N \implies  \beta - \epsilon < x_n < \beta + \epsilon.$$ 
By the construction of $\{s'_n\}$, the follwoing inequalities hold: 
$$\beta + \epsilon > x_N = s'_{m_1 + k_1 + \cdots + m_{N-1} + k_{N-1} + m_N} > s'_{m_1 + k_1 + \cdots + m_{N-1} + k_{N-1} + m_N + 1} \\> \cdots > s'_{m_1 + k_1 + \cdots + m_{N-1} + k_{N-1} + m_N + k_N} = y_N < s'_{m_1 + k_1 + \cdots + m_{N-1} + k_{N-1} + m_N + k_N+1} < \cdots < s'_{m_1 + k_1 + \cdots + m_{N-1} + k_{N-1} + m_N + k_N+(m_{N+1}-1)} < s'_{m_1 + k_1 + \cdots + m_{N-1} + k_{N-1} + m_N + k_N+m_{N+1}} = x_{N+1} < \beta + \epsilon.$$ 
Let $M := m_1 + k_1 + \cdots + m_{N-1} + k_{N-1} + m_N$.
The above inequalities say that $s'_n < \beta + \epsilon$ for all $n \in \{M, M+1, \cdots, M + k_N + m_{N+1}\}$.  
And we see that the following ineqality holds:  
$s'_n < \beta + \epsilon$ for all $n \in \{M, M+1, \cdots\}$.  
