Weakly null sequence in Schreier space Define the Schreier space $X:=\overline{c_{00}}^{\Vert\cdot\Vert}$ where $$\Vert x\Vert = \sup\bigg\{\sum_{i=1}^k|x_{n_i}|:k\le n_1<n_2\cdots <n_k\bigg\}.$$
Show that there exists a weakly null sequence $(x_n)\in X$ such that for any subsequence $(x_{n_m})$, the Cesaro sum $\frac{1}{m}\sum_{i=1}^m x_{n_i}\not\rightarrow 0$ as $m\rightarrow\infty$.
I know that $x_n\rightharpoonup 0$ iff $x_n^{(k)}\rightarrow 0$ as $n\rightarrow \infty $ since $c_{00}$ is dense in $X$. I've tried many different sequences with componentwise decay e.g. shifts of $(1,\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{4},\frac{1}{4},\frac{1}{4},\frac{1}{4},\cdots)$ but none of them seem to work. Does anyone have an example?

Let $(e_n)$ be the standard sequence of Schauder basis elements. I.e $e_n(j)=\delta_n^j$, with $\delta$ the Kronecker-Delta function. Clearly this is a weakly null sequence, as pointwise it tends to $0$. Choose any subsequence $(e_{n_k})$. If we let $\lfloor x\rfloor$ denote the floor of any real number, then for any natural $m>1$ we know that
$$\|\frac{1}{m}\sum_{k=1}^me_{n_k}\|\geq\frac{1}{m}\sum_{j=1}^{\lfloor m/2\rfloor}|e_{n_{j+\lfloor m/2\rfloor}}(n_{j+\lfloor m/2\rfloor})|\geq 1/3,$$
which means that the Caesaro sums of $e_{n_k}$ cannot converge to $0$.
