On p.98 of these notes, or the first result to come up for the search "Goldstine" one finds a proof of theorem 7.24.
http://www.math.uwaterloo.ca/~lwmarcou/Preprints/LinearAnalysis.pdf
I don't understand the step using Hahn-Banach. Specifically, where the linear functional is chosen as an evaluation functional at some point in $X^*$. Normally, he's only guaranteed some linear functional in $X^{***}$ that is weak* continuous in the weak* topology on $X^{**}$. Is there some reason why the evaluation functionals make up all of these? If this is not what's going on please let me know, or if the notes are wrong, please suggest an alternative proof. Thanks!