Is there any flaw in following proof
Theorem : consider a bounded sequence $(a_n)$ and let $s_k = \sup \{|\frac{a_{n+1}}{a_{n}}| : n \geq k\}$ and $l = \lim_{k \rightarrow \infty} s_k$ $\,\,\,\,$ if $l < 1$ then $\sum_{n=1}^{\infty} a_n$ absolutely converges
proof
- since consider $l'$ st $0<l<l'<1$
- now fix $\epsilon = |l'-l|$
- we know that $\exists N \in \mathbb{N} \, , \, \forall n \geq N \,\,\bigg| s_{n} - l \bigg| < \epsilon $ which implies $ s_{n} < l'$
- fix at $N$ from which we have that $s_{N} < l' \implies |\frac{a_{n+1}}{a_{n}}| < l' \,\,\, \forall n \geq N$
- $|a_{n+1}| < l'|a_{n}| \,\,\, \forall n \geq N$ from which we have that $a_{m} < (l')^{m-N}|a_{N}|$ $\,\,\,\forall m > N$
- $\sum_{k=N+1}^{\infty} |a_{n}| \leq \sum_{k=1}^{\infty} (l')^k |a_{N}| = |a_{N}| \sum_{k=1}^{\infty} (l')^k < \infty$
- hence $\sum_{n=1}^{\infty}|a_n|$ absolutely converges