I'm solving an exercise from Lax's Functional analysis. The section concerns generalized limits (more particularly, Banach limits), which are obtained by applying the Hahn-Banach theorem to the classical limit functional on $\ell^\infty$.
The exercise wants me to construct a Banach limit which agrees with the Cesaro limit of Cesaro summable sequences, that is, bounded sequences such that "the arithmetic means of the partial sums converge."
If I understand this final sentence correctly (and wiki seems to confirm this), a sequence $(x_1,x_2,\ldots)$ is Cesaro summable if $$ \lim_{n\to\infty}\frac{1}{n}\sum_{k=1}^n s_k < \infty, \quad\text{where}\quad s_k = \sum_{j=1}^k x_j. $$ On the other hand, in this post, another definition of Cesaro summability is used, namely there it is required that $$ \lim_{n\to\infty} \frac{1}{n}\sum_{k=1}^n x_k < \infty. $$
My problem is that I would like to dominate the Cesaro limit functional by the (subadditive and positive homogeneous) function $$ p(x) = \limsup_{n\to\infty} x_n, $$ which Lax uses to extend of the classical limit. This is easily done if we use the latter definition, but I'm not sure how to do it (or if it's even possible) if we use the former definition.
My other idea was to use $$ p(x) = \limsup_{n\to\infty} |s_n|, $$ but this is not even well-defined since we are dealing with general bounded sequences.
How would you interpret the problem, i.e. which definition of Cesaro summability am I supposed to use? What is a good candidate for $p$ if we use the former definition?
