Using Cauchy convergence criterion to prove that, "if convergent series contains only finitely many negative terms then it is absolutely convergent"

Question

This question is asked already here Proof verification: convergent series with a finite number of negative terms is Absolutely Convergent

But, answer to this question used different method (it does not explain how the Cauchy convergence criterion applied)

Let $s_n$ be nth partial sum of $\sum a_n$ and $t_n$ be nth partial sum of $\sum |a_n|$ and let $a_n≥0$ for all $n>K$ then,

if $m>n>K$ we have, $t_m-t_n= s_m-s_n$

Now, how to apply Cauchy convergence criterion to establish the convergence of $(t_n)$ ?

I know, by Cauchy convergence criterion we have, for given $\epsilon >0$ there exists $M(\epsilon)\in\mathbb{N}$ such that, if $m>n>M(\epsilon)$ then $|s_m-s_n|=|s_{n+1}+s_{n+2}+...+s_m|<\epsilon$

(But then, why should be this $M(\epsilon)>K$?)

how to proceed? Please help

Answer 1

Let $N(\epsilon)$ be an integer greater than both $M(\epsilon)$ and $K$. Then for all $m>n>N(\epsilon)$, $a_n, \ldots, a_m$ are all positive, and their sum is smaller than $\epsilon$. You may apply Cauchy’s criterion from here.