Let $e_n:=δ_{kn}$, for $k\in\mathbb N$. Given the sequence $(a_n):=\sum\limits_{k=n}^∞e_k\subset\ell^\infty$, i.e.$$((1,1,\cdots),(0,1,1,\cdots),(0,0,1,1,\cdots),\cdots).$$ I want to know if $(a_n)$ converges weakly to zero.
Assume $(a_n)$ does not converges weakly, then I could use Hahn-Banach to find a linear functional $\varphi \in (\ell^\infty)^*$ with $$ \lim_{n \to \infty} \varphi((a_n)) \neq 0 \; , $$ but if $(a_n)$ converges weakly to zero, then I have no idea to show this, since I lack an usefull characterization of $(\ell^\infty)^*$.
It is possible to use Banach limits. A Banach limit $\phi$ is in particular a linear continuous functional on $\ell^\infty$ such that if $x\in \ell^\infty$ converges to some $\ell$, then $\phi(x)=\ell$. Here $\phi(a_n)=1$ hence there cannot be weak convergence to zero.
Actually, there is no weak convergence at all since using the evaluation maps $e_n\colon (x_k)_{k\geqslant 1}\in\ell^\infty\mapsto x_n$, the only potential weak limit is the null sequence.