Let $(x_{n})_{n}$ be a sequence in a Banach space $X$. Assume that the set $\{x_{n}:n=1,2,\cdots\}$ is finite. Let $(f_{m})_{m}$ be a weak*-null sequence in $X^{*}$ with $\|f_{m}\|\leq 1$($m=1,2,\cdots$) satisfying the following conditions:

(1) the limit $a_{m}:=\lim\limits_{n}\langle f_{m},x_{n}\rangle$ exists for each $m$;

(2) the limit $a:=\lim\limits_{m}a_{m}$ exists.

Question. $a=0$ ?

Thank you !

Let $(x_{n})_{n}$ be a sequence in a Banach space $X$. Assume that the set $\{x_{n}:n=1,2,\cdots\}$ is finite. Let $(f_{m})_{m}$ be a weak*-null sequence in $X^{*}$ satisfying the following conditions:

(1) the limit $a_{m}:=\lim\limits_{n}\langle f_{m},x_{n}\rangle$ exists for each $m$;

(2) the limit $a:=\lim\limits_{m}a_{m}$ exists.

Question. $a=0$ ?

Thank you !