I am studying Laurent series from A Pathway to Complex Analysis by S.Kumaresan.The chapter begins with a simple definition.An expression $\sum\limits_{n=-\infty}^{\infty} a_n$ is said to converge to $s\in \mathbb C$ if given $\epsilon>0$ there exists $N\in \mathbb N$ such that $|(a_{-m}+...+a_{n})-s|<\epsilon$ for all $m,n\geq N$.
Now there is a statement which says that:
$\sum\limits_{n=-\infty}^{\infty}a_n$ is convergent to $s$ $\implies \sum\limits_{n=1}^\infty a_{-n}$ and $\sum\limits_{n=0}^\infty a_n$ are convergent to $s_1$ and $s_2$ respectively satisfying $s=s_1+s_2$.
I am trying to prove this result.I thought of introducing the idea of Cauchy convergence but that version does not seem to fit properly here.How can I show that this statement is true directly from definition?
Taking $m=N$ we get $|(a_{-N}+...+a_{n})-s|<\epsilon$ for all $n\geq N$. This gives convergence of $ \sum\limits_{k=-N}^{\infty} a_k$ (since its partial sums form a Cauchy sequence) . Convergence of a series is not altered by removing/adding/altering a finite number of terms. Omitting the terms $a_{-N}, a_{-N+1},...,a_{-1}$ we see that $ \sum\limits_{k=0}^{\infty} a_k$ is convergent .