Let $a_n$ be a non-negative sequence, so that $\lim \limits_{n \to \infty}b_n= \frac {a_1+a_2+…+a_n} n=0$.
Prove that there is a subsequence $a_{n_k}$ that converges to $0$.
We know that since $\lim \limits_{n \to \infty}b_n=0$, then every subsequence of $b_n$, say $b_{n_k}$ also converges to $0$. Then, which means that we have $\infty$ members of $b_{n_k}$ in $(-\epsilon$, $\epsilon$), but since all the numbers are non-negative, then all of them must be in [$0$, $\epsilon$). This is where I'm stuck, any ideas?
Best Answer
You must show that
$$\forall \epsilon > 0,\,\forall n \in \Bbb N,\, \exists m\geqslant n,\, a_m < \epsilon.$$
Suppose that this were not true, that is:
$$\exists \epsilon > 0,\,\exists n \in \Bbb N,\, \forall m\geqslant n,\, a_m \geqslant \epsilon.$$
In other words, from some $n$ onwards, all the $a_m$ are $\geqslant \epsilon$. What could you say about $b_n$ in this case?