I'm preparing for my exam in functional analysis and I'm trying to solve the following question:
Let X be a Banach space, let E$\subset$X be dense and let $(f_n)$ be a sequence in $X^*$*. Prove that $f_n \rightarrow f$ in the weak$^*$-sense if and only if $(||f_n||)$ is bounded and $(f_n(x))$ is Cauchy in $\mathbb{K}$ for all $x\in E$.
I have proven the ($\rightarrow$) direction, but I don't know how to do the ($\leftarrow$) direction. I know I need to show that $f_n(x) \rightarrow f(x)$ for all $x\in X$ so I'm trying to apply the triangle inequality in some way but I can't seem to get it work. Thank you in advance!