Here's Prob. 14, Chap. 3, in the book Principles of Mathematical Analysis by Walter Rudin, 3rd edition:
If $\left\{ s_n \right\}$ is a complex sequence, define its arithmetic mean $\sigma_n$ by $$ \sigma_n = \frac{s_0 + s_1 + \cdots + s_n}{n+1} \ \ \ (n = 0, 1, 2, \ldots). $$
(a) If $\lim s_n = s$, prove that $\lim \ \sigma_n = s$.
(b) Construct a sequence $\left\{ s_n \right\}$ which does not converge, although $\lim \ \sigma_n = 0$.
(c) Can it happen that $s_n > 0$ for all $n$ and that $\lim \sup s_n = \infty$, although $\lim \ \sigma_n = 0$?
(d) Put $a_n = s_n - s_{n-1}$, for $n \geq 1$. Show that $$s_n - \sigma_n = \frac{1}{n+1} \sum_{k=1}^n k a_k.$$ Assume that $\lim \left( n a_n \right) = 0$ and that $\left\{ \sigma_n \right\}$ converges. Prove that $\left\{ s_n \right\}$ converges. [This gives a converse of (a), but under the aditional assumption that $n a_n \to 0$. ]
(e) Derive the last condition from a weaker hypothesis: Assume $M < \infty$, $\left\vert n a_n \right\vert \leq M$ for all $n$, and $\lim \ \sigma_n = \sigma$. Prove that $\lim s_n = \sigma$, by completing the following outline:
If $m < n$, then $$s_n - \sigma_n = \frac{m+1}{n-m} \left( \sigma_n - \sigma_m \right) + \frac{1}{n-m} \sum_{i=m+1}^n \left( s_n - s_i \right).$$ For these $i$, $$ \left\vert s_n - s_i \right\vert \leq \frac{(n-i)M}{i+1} \leq \frac{(n-m-1)M}{m+2}.$$ Fix $\varepsilon > 0$ and associate with each $n$ the integer $m$ that satisfies $$ m \leq \frac{n-\varepsilon}{1+ \varepsilon} < m +1.$$ Then $\frac{m+1}{n-m} \leq \frac{1}{\varepsilon}$ and $\left\vert s_n - s_i \right\vert < M \varepsilon$. Hence $$ \lim_{n\to\infty} \sup \left\vert s_n - \sigma \right\vert \leq M \varepsilon.$$ Since $\varepsilon$ was arbitrary, $\lim s_n = \sigma$.
My effort:
Part (a):
If $\lim s_n = s$, then we can find a natural number $N$ such that $n > N$ implies that $$ \left\vert s_n - s \right\vert < 1.$$ So, for $n > N$, we have \begin{align} \left\vert \sigma_n - s \right\vert &= \left\vert \frac{ s_0 + s_1 + \cdots + s_n}{n +1} - s \right\vert \\ &\leq \frac{1}{n+1} \left( \left\vert s_0 - s \right\vert + \cdots + \left\vert s_N - s \right\vert + \cdots + \left\vert s_n - s \right\vert \right) \\ &\leq \frac{1}{n+1} \left( (N+1) \max \left\{ \left\vert s_0 - s \right\vert, \ldots, \left\vert s_N - s \right\vert \right\} + \left\vert s_{N+1} - s \right\vert + \cdots + \left\vert s_n - s \right\vert \right) \\ &< \frac{1}{n+1} \left( (N+1) \max \left\{ \left\vert s_0 - s \right\vert, \ldots, \left\vert s_N - s \right\vert \right\} + (n-N) \right) \end{align} Let $\varepsilon > 0$ be given. What next?
Part (b):
Let $s_n = (-1){n+1}$ for $n = 0, 1, 2, \ldots$. Then $$ \sigma_n = \begin{cases} \frac{1}{n+1} \ \mbox{ if } n \mbox{ is even}; \\ 0 \ \mbox{ if } n \mbox{ is odd}. \end{cases} $$ Then $\left\{ s_n \right\}$ fails to converge, but $\lim \ \sigma_n = 0$. Is this example correct?
Part (c):
My feeling is the answer is no, but I cannot establish this rigorously. How to?
Part (d):
If we put $a_n = s_n - s_{n-1}$, for $n \geq 1$, then $s_n = a_0 + s_1 + \cdots + a_n$, and so \begin{align} s_n - \sigma_n &= s_n - \frac{ s_0 + s_1 + \cdots + s_n}{n+1} \\ &= \frac{ (n+1) s_n - s_0 - s_1 - \cdots - s_n }{n+1} \\ &= \frac{ (n+1) \left( s_0 + a_1 + \cdots + a_n \right) - s_0 - \left( s_0 + a_1 \right) - \left( s_0 + a_1 + a_2 \right) - \cdot - \left( s_0 + a_1 + \cdots + a_n \right) }{n+1} \\ &= \frac{ a_1 + 2 a_2 + \cdots + n a_n }{n+1}. \end{align} Now we assume that $\lim_{n \to \infty} n a_n = 0$ and that $\left\{ \sigma_n \right\}$ converges. How to show that $\left\{ s_n \right\}$ convreges?
Part (e):
If $m < n$, then \begin{align} & \frac{ m+1 }{ n-m } \left( \sigma_n - \sigma_m \right) + \frac{1}{n-m} \sum_{i = m+1}^n \left( s_n - s_i \right) \\ &= \frac{ m+1 }{ n-m } \left( \frac{ s_0 + s_1 + \cdots + s_n}{n+1} - \frac{ s_0 + s_1 + \cdots + s_m}{m+1} \right) + \frac{1}{n-m} \sum_{i = m+1}^n \left( s_n - s_i \right) \\ &= s_n + \frac{1}{n-m} \left[ (m+1) \frac{ s_0 + \cdots + s_n}{n+1} - \left( s_0 + \cdots + s_n \right) \right] \\ &= s_n - \sigma_n. \end{align} How to proceed from here?
