Here is Abbott's proof:
Assume $\sum\limits_{k = 1}^{\infty} a_k$ converges absolutely to $A$, and let $\sum\limits_{k = 1}^{\infty} b_k$ be a rearrangement of $\sum\limits_{k = 1}^{\infty} a_k$. Let's use $s_n$ to denote the partial sums of the original series and $t_m$ for the partial sums of the rearranged series. Thus, we want to show that $(t_m) \to A$.
Let $\epsilon > 0$. By hypothesis, $(s_n) \to A$, so choose $N_1$ such that $|s_n - A| < \frac{\epsilon}{2}$ for all $n \geq N_1$. Because the convergence is absolute, we can choose $N_2$ so that $\sum\limits_{k = m + 1}^{n} |a_k| < \frac{\epsilon}{2}$ for all $n > m \geq N_2$. Now take $N = max \{N_1, N_2\}$. We know that the finite set of terms $\{a_1, \ldots, a_N\}$ must all appear in the rearranged series, and we want to move far enough out in the series $\sum\limits_{n = 1}^{\infty} b_n$ so that we have included all of these terms. Thus, choose $M = max \{f(k) : 1 \leq k \leq N \}$.
It should now be evident that if $m \geq M$, then $(t_m - s_N)$ consists of a finite set of terms, the absolute values of which appear in the tail $\sum\limits_{k = N + 1}^{\infty} |a_k|$. Our choise of $N_2$ earlier then guarantees $|t_m - s_N| < \frac{\epsilon}{2}$, and so $|t_m - A| = |t_m - s_N + s_N - A| \leq |t_m - s_N| + |s_N - A| < \frac{\epsilon}{2} + \frac{\epsilon}{2} = \epsilon$ whenever $m \geq M$
Now my question: if $t_m - S_N$ is just a bunch of terms $a_k$ where $k \geq N+1$, then why is it not enough to just use the fact that $\sum a_k$ is Cauchy since it congers, why couldn't I pick $N_2$ such that $|a_{m+1} + \ldots + a_n|$ for all $n > m \geq N_2$? I know that using absolute convergence is necessary, but what is wrong with my previous logic?
You can show that $(t_m - s_N)$ is the sum of a finite set of terms from the series, but it is not necessarily true that this is a set of consecutive terms from the series. It could be (and will be, if $a_n$ is a conditionally convergent series and $t_m$ has been arranged to come out to a different sum) that there are infinitely many cases where the terms that sum to $(t_m - s_N)$ are not consecutive terms of $a_n$, that $(t_m - s_N)$ is not equal to any sum of the form $a_k + \ldots + a_n$, and therefore it is irrelevant what the value of $|a_{m+1} + \ldots + a_n|$ is for any $n$.