Does bounded $\{ \frac{1}{n}\sum_{k=1}^{n}a_k \}_n$ imply $\frac{a_n}{n} \rightarrow 0$?

Question

Let $\{ a_n\}_n$ be a positive sequence. If the sequence $p_n = \frac{1}{n}\sum_{k=1}^{n}a_k$ is bounded, then does this imply $\limsup_{n \rightarrow \infty} \frac{a_n}{n} = 0$?

Answer 1

Set $$a_n=\begin{cases}n & n=10^k\text{ for some }k\in\mathbb{N}\\0 & \text{otherwise}\end{cases}$$

Now the lim sup of $\frac{a_n}{n}$ is $1$, and we have $p_n=\frac{11111\cdots 1}{10000\cdots 0}=1.1111\cdots 1$ (for $n$ a power of $10$), which is bounded above by $\frac{10}{9}$. For $n$ not a power of $10$, $p_n$ is even smaller, so also bounded above by $\frac{10}{9}$.