I'm self-studying from the book Understanding Analysis by Stephen Abbott and I have a question about the proof of the Absolute Convergence Test (theorem 2.7.6 on page 65).
The author states that since the series $\sum_{n=1}^\infty |a_n|$ converges, we know that, given $\epsilon >0$, there exists an $N \in \mathbb{N}$ such that: \begin{equation} |a_{m+1}| + |a_{m+2}| + \cdots + |a_n| < \epsilon \end{equation} for all $n > m \geq N$.
My question is: why this is true? I don't think the author has proved this anywhere in the book.
Intuitively, I can sort of understand the above statement only if the "tail" of the series becomes increasingly small (which naively seems to a property of the convergent series, but I'm not sure if this is always true). But this all seems very "handwavy". Can anybody shed some light on my confusion?
You're assuming the series $\sum\limits_{n=1}^\infty |a_n|$ converges. This, by definition means that the sequence of partial sums, $(S_m)$, given by $S_m=\sum\limits_{n=1}^m |a_n|$, converges.
If you look closely, you should be able to see that your condition is just saying $(S_m)$ is a Cauchy sequence (which it is, of course).