Say you have a sequence $\{a_n\}_{n=1}^\infty$ that has a sequence of averages in the form $\{b_n\}_{n=1}^\infty$ such that for all natural numbers $n$, $$b_n = \frac{1}{n}\sum_{i=1}^{n} a_i.$$
I'm trying to figure out several examples of convergence and divergence in this setup.
(1) An example where $a_n$ does not converge but $b_n$ converges.
My thought for this is to use the sequence $(-1)^n$ for $a_n$, for which the limit of the sequence of averages would approach $0$, indicating convergence.
(2) An example where $a_n$ is bounded but $b_n$ does not converge.
My thought would be to use a sequence of $0$s and $1$s for $a_n$ such that the sequence of averages $b_n$ oscillates between $0$ and $1$. Not sure how to demonstrate the sequence $a_n$ is bounded in this case.
