A sequence $x_n$ is almost convergent if $$ \exists L\in\mathbb R: \forall\varepsilon\in\mathbb R: \exists p_0\in\mathbb N: \forall p>p_0: \forall n\in\mathbb N: \left| L-\frac1p\sum_{i=n}^{n+p-1}x_i \right| \leq \varepsilon \\ $$ The condition can be rewritten as $$ p(L-\varepsilon) \leq \sum_{i=n}^{n+p-1}x_i \leq p(L+\varepsilon) $$ and with that it seems obvious that the sum of two almost convergent sequences with limits $L_1$ and $L_2$ will almost converge to $L_1+L_2$.
On the other hand, if we consider the product (element-wise) of two sequences, things are more complicated. If we take for example $x_n = y_n = (-1)^n$, we can prove that both almost converge to 0, on the other hand $x_n \cdot y_n = 1$. So my question is: can we define the product in a different way so that the result is a sequence that almost converges to the product of the limits of the input sequences?
To provide some more context, I am nostly interested in almost convergent sequences of integers and am lookin for a form of product that would be also a sequence of integers. The larger problem I am trying to solve is to define the reals almost convergent sequences of integers.
Update: Based on some simple examples that I did by hand, it seems that the following product could be a candidate: $$ c_n = \frac1{n+1}\sum_{i=0}^n a_i \cdot b_{n-i} $$ I am however unable so far to prove that this product is almost convergent whenver $a_n$ and $b_n$ are.