On the Weak* Convergence of Subdifferentials of Convex Functions

Let us assume that a sequence $\{ f_{n} \}_{n=1}^{\infty }$ of proper lower semicontinuous convex functions is bounded on some open subset of a weakly compactly generated Banach space. It is shown that if $\{ f_{n} \}_{n=1}^{\infty }$ is a Mosco converging sequence, then for every subgradient $x^*$ of $f$ at $x$ there are subgradients $x^{*}_{n}\in \partial f_{n}(x_{n})$ such that $\{ x^{*}_{n} \}_{n=1}^{\infty }$ is weakly$^*$ converging to $x^*$.