# Positive and negative portions of conditionally convergent series

I want to prove that for a conditionally convergent series $\sum_{n = 1}^{\infty} a_n$, that its positive subseries (Let $K := \{k \vert a_k > 0\}$, and the positive subseries is $\sum_{j \in K} a_j$) and a (similarly defined) negative subseries diverge.

I know that there is a proof here but I'd want to prove it moreso using definitions. Call $s_n$ the sum of the first n $a_i$, and we know $\exists s$ so for all $n> N$, $\vert s_n - s \vert < \epsilon$. Call $\bar{s}_n$ the sum of the first n $\vert a_i \vert$, and we know $\forall M \exists n$ so $\bar{s_n} >M$.

So since the series $a_n$ is conditionally convergent, there can exist no $N$ so that after that $N$, all terms have the same sign. Thus I want to say something along the lines of, given some $\epsilon$, say 1, there must be a positive subseries of $a_n$ such that $\sum_{i=1}^{n} a_{k_i}$ is above the interval $(s-1,s+1)$, then we can pick do this again with more positive terms so its above the interval $(s-2,s+2)$ and so on. Can I go anywhere with this argument?