We have a non-negative sequence $\{a_n\},~n\in\mathbb{N}$ and its arithmetic mean $\frac{\sum_{i=0}^{N-1}a_i}{N}$ goes to 0 as $N\to\infty$. Can we claim that in this case, there exists a subsequence $\{a_{k_n}\}$ such that $\lim_{n\to\infty}a_{k_n}=0$?
I have a proof of yes as follows:
We prove it by contradiction. Suppose that such a subsequence does not exist. Then there exists an integer $N$ and positive constant $\epsilon$ such that $\forall n\ge N$, $a_n\ge\epsilon$. However, in this case, $\lim_{N\to\infty}\frac{\sum_{i=0}^{N-1}a_i}{N}=\epsilon>0$, which contradicts the fact that $\lim_{N\to\infty}\frac{\sum_{i=0}^{N-1}a_i}{N}=0$.
My question is: “Suppose that such a subsequence does not exist. Then there exists an integer $N$ and positive constant $\epsilon$ such that $\forall n\ge N$, $a_n\ge\epsilon$.” Is this statement correct?