We know that the Cauchy Criterion of a series is as follow (proof is taken as excerpt from an analysis book):
Theorem: A series $\sum_{j=1}^{\infty}a_j$ converges iff for all $\epsilon>0$ there is an $N\in \mathbb{N}$ so that for all $n\ge m \ge N$ we have $|\sum_{j=m}^{n} a_j|< \epsilon$.
Let $s_n$ denotes partial sum $\sum_{j=1}^n a_j$.
We know that the proof of $\Rightarrow$ direction makes use of the fact that convergent series implies the sequence of partial sum converges and thus is Cauchy in $\mathbb{R}$. So here is part of the proof, since sequence of partial sum is Cauchy, therefore given $\epsilon>0$, there exists $N \in \mathbb{N}$ such that for all $n \ge m \ge N$, $|s_n - s_{m-1}|< \epsilon$.
I do not get the $s_{m-1}$ part, i.e. I do not get why you can pick $m-1$, since there is possibility for $m=N$, and thus having $m-1 When I was thinking about this, I was rather confused, and my original thought was following: