if $F$ is a Banach space and $f_n \subset F^* $ weak star convergent to $f\in F^*$. If further $x\in F$ is the weak limit of $(x_n)_n \subset F$ does then $f_n(x_n) \longrightarrow f(x)$ hold?

We know that for all $n$: $\lim_m f_n(x_m) = f_n(x)$ and for all $m$: $\lim_n f_n(x_m) = f(x_m)$ so what can I infer about $\lim_n\lim_m f_n(x_m)$? I thought as the limit of $f_n$ is again in $F^*$ i could just put $\lim_n(\lim_m f_n(x_m)) = \lim_n f_n(x)$?!