Two convergent series, Cauchy Product I am trying to prove the this Theorem :
If $\sum{(a_n)}$ coverges to A, $\sum{(b_n)}$ converges to B, let $\sum{(c_n)}$ be their Cauchy product, then 
$$\frac{1}{n} (C_1+C_2+\dots+C_n)\rightarrow AB$$
Let $A_n$, $B_n$, $C_n$ be the partial sums of $\sum{a_n}$, $\sum{b_n}$, $\sum{c_n}$ respectively.
The book gives a hint: $$C_1+\dots+C_n=A_1B_n+\dots+A_nB_1$$
I have no idea why this equality is true, and where to use the convergence of these two series. Can someone help me?
 A: Here is a technique to manipulate these sums.
We will need the indicator function 
$$1_{(k \leqslant j)} = \begin{cases}1 \,\,\text{if}\,\, k \leqslant j \\ 0 \,\,\text{if}\,\, k > j\end{cases}$$
The Cauchy product is
$$C_p = \sum_{j=1}^p \sum_{k = 1}^ja_kb_{j-k+1} \\ = \sum_{j=1}^p \sum_{k = 1}^pa_kb_{j-k+1}1_{(k \leqslant j)} \\ = \sum_{k=1}^p a_k\sum_{j = 1}^pb_{j-k+1}1_{(k \leqslant j)} \\ =\sum_{k=1}^p a_k\sum_{j = k}^pb_{j-k+1} \\ =\sum_{k=1}^p a_k\sum_{j = 1}^{p-k+1}b_{j} \\ = \sum_{k=1}^p a_kB_{p-k+1} \\ =a_1B_p + a_2B_{p-1} + \ldots + a_p B_1.$$
Using a similar approach you can now show
$$C_1+\dots+C_n=A_1B_n+\dots+A_nB_1.$$
To compute the limit, note that
$$\frac1{n}\sum_{k=1}^n C_k = \frac1{n}\sum_{k=1}^m C_k + \frac1{n}\sum_{k=m+1}^{n-m} A_kB_{n-k} + \frac1{n}\sum_{k=n-m+1}^n C_k.$$
For any $\epsilon > 0$ there exists $m \in \mathbb{N}$ such that if $k > m$ we have $A - \epsilon < A_k < A + \epsilon$ and $B - \epsilon < B_k < B + \epsilon.$
With $m$ fixed, the first and third sums on the RHS have a finite number of terms and converge to $0$ as $n \to \infty$.  The middle sum has $n - 2m$ terms and is bounded between $(A- \epsilon)(B - \epsilon)(1 - 2m/n)$ and $(A+ \epsilon)(B + \epsilon)(1 - 2m/n).$ Hence, $\limsup$ and $\liminf$ are squeezed between $(A- \epsilon)(B - \epsilon)$ and $(A+ \epsilon)(B + \epsilon)$ as $n \to \infty$, and, therefore the LHS converges to $AB$.
A: Actually that's not quite right.  You want $$C_1 +... + C_{n+1} = A_1B_n + \ldots + A_n B_1$$
You can verify it by counting how many times a given $a_i b_j$ appears in each term.
As for the limit, note that if $A_i \to A$ and $B_i \to B$, nearly all of the terms on the right will be close to $AB$...
