The problem goes as:
Let $X$ be a Banach space and let $(f_n)^{\infty}_{n=1}$ be a weak$^*$ Cauchy sequence in $X$, that is, $(f_n(x))^{\infty}_{n=1}$ is a Cauchy sequence $\forall x \in X$. Show that $(f_n)^{\infty}_{n=1}$ is weak$^*$ convergent in $X'$ (dual space of $X$).
My idea was to find a proper weak$^*$ limit and then apply the Banach-Steinhaus theorem to conclude that $(f_n)^{\infty}_{n=1} \xrightarrow{w^*} f \in X'$. But I haven't been able to accomplish this.
Appreciating all help.