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?