Does average value always exist? If $a_i \in {\mathbb R}_+$ for all $i \in {\mathbb Z_+}$, and
\begin{align}
\limsup_{n \rightarrow \infty} \frac{1}{n} \sum_{i=1}^{n} a_i < \infty,
\end{align}
then does the $\lim$ exist? In other words, is there any $c \in {\mathbb R}$ such that
\begin{align}
\lim_{n \rightarrow \infty} \frac{1}{n} \sum_{i=1}^{n} a_i =c.
\end{align}
Notes: $(1/n)\sum_{i=1}^{n}a_i$ does not increase with $n$, even though $\sum_{i=1}^{n}a_i$ does. Hence, the Monotone Convergence Theorem cannot be applied.

No, there are plenty of examples where this fails. Let $a_i=1$ whenever $i$ does not have a 1 in it's decimal expansion. You'll see plenty of such $i$ for $i<1000$ but then suddenly no $i$ in $1000<i<2000$. Then it will jump again. The limsup exists since it is bounded above by $1$. But the average will oscillate forever. To prove this, notice that the average drops below $1/2$ whenever $i\in [10^x,2\cdot 10^x]$ but then is above $1/2$ when $i\in [2\cdot 10^x,3\cdot 10^x]$.
