Question about convergence of sub-gradients

Question

Suppose, {f_n} form a sequence of convex functions. They are not necessarily differentiable. {f_n} uniformly converge to a function f. I want to know whether at any point x_0, for any sub-gradient v ∈ ∂f(x_0), there must exist at least one sequence of {v_n} such that v_n ∈ ∂f_n(x_0) for all n’s and lim v_n = v?

Answer 1

No, take the functions $f_n \colon \mathbb R \to \mathbb R$, $$ f_n(x) = |x + 1/n|.$$