TITLE: sequence and series (complex analysis)
QUESTION [5 upvotes]: Let $N_0\in \mathbb{N}.$ If a sequence of complex numbers $\{F_N\}_{N \in \mathbb{N}}$ has the following properties:
$$\lim_{N \rightarrow \infty} |F_N|^{1/N}=0$$ and for all $N \geq N_0$,
$$|F_N|\leq \sum_{k=N+1}^{\infty} |F_k|,\quad \quad \quad  $$
then there exists $N_1\in \mathbb{N}$ such that $F_N=0$ for all $N \geq N_1.$
I found this argument on a paper but I cannot prove it. Could you please help me or give me an idea? Thank you.
Masik

REPLY [3 votes]: This may be another way to do it. 
So given all the conditions, there exists $M$ such that $\forall N\ge M, |F_N|^{1/N}<1/3,$ and $|F_N|\le\sum_{k=N+1}^\infty|F_k|,$ I claim that for those $N$'s, $|F_N|\le(1/2)^n(1/3)^N$ for any non-negative integer $n$. Then by letting $n$ approaches $\infty,$ we actually force $|F_N|=0.$
To prove the claim, we do induction on $n.$ When $n=0,$ it follows immediately from $|F_N|^{1/N}<1/3.$ In general, by induction hypothesis 
$$|F_N|\le\sum_{k=N+1}^\infty|F_k|\le\sum_{k=N+1}^{\infty}(1/2)^n(1/3)^k=(1/2)^{n+1}(1/3)^N$$