Given a Banach space $X$ . Let $F\in X^{**}$ . We know that closed unit ball $B(X)$ is w*-dense in $B(X^{**})$ due to Goldstein's theorem . In paper
Rodríguez Palacios, Angel, A note on Arens regularity, Q. J. Math., Oxf. II. Ser. 38, 91-93 (1987). ZBL0617.46053.
they mention that together with Mackey-Arens theorem and Goldstein's theorem, we can find a bounded net $(x_\alpha)$ in $X$ such that it converges to $F$ in Mackey topology $\tau(X^{**},X^*)$.
Its direct consequence of Goldstein's theorem to find a bounded net convergent to $F$ in w*-topology. But I cannot think how to construct one which is convergent in Mackey topology $\tau(X^{**},X^*)$.
The Mackey-Arens theorem says that the Mackey topology $\tau(X^{**},X^*)$ is a (in fact, the finest)locally convex topology on $X^{**}$ so that the dual of $(X^{**},\tau(X^{**},X^*))$ is $X^*$ (more precisely, consist only of evaluations in points of $X^*$) which also holds for $(X^{**},\sigma(X^{**},X^*))$. The Hahn-Banach theorem thus implies that the closures of convex sets in both spaces coincide so that $B(X)$ is Mackey-dense in $B(X^{**})$.