show that $f_n \xrightarrow{w^*} 0$ in $l^\infty$.

Question

let $f_n$ be sequence of unit vectors in $l^p,1\leq p\leq \infty$.

then show that $f_n \xrightarrow{w^*} 0$ in $l^\infty$.

i know that if $X$ is any normed space , and $f_n ,f\in X^*$ then $f_n\xrightarrow{w^*} f$ if $f_n(x)\to f(x) \forall x\ \in X$ .

and i also know $(l^1)^*=l^{\infty}$. it mean for every $f\in (l^1)^*$ there exist sequence $a_n \in l^{\infty}$ such that $$f(x)=\sum _{i=1}^{\infty} \alpha_i a_i$$ where $(\alpha_i)$ is coordinates of $x$ with respect to basis $(e_i)$. also $||f||=||a_n||_{\infty}$

now how to use this in this problem. any hint

thanks in advanced