Consider the $l$-$0$ minimization problem: $$\min_{x} f(x) + \lambda \|x\|_0$$ First, prove that $g(x) = \|x\|_0$ is lower semicontinuous and then find the limiting subdifferential $\partial g(x)$ and horizon subdifferential $\partial^{\infty} g(x)$. Next, given that $f$ is lower semicontinuous and it's horizon subdifferential $\partial^{\infty} f(x) = \{0\} \forall x$, find the first-order necessary optimality condition for a point $x^*$ to be a minimizer of the given problem.
I know I need to show that $$\liminf_{y \rightarrow x} \|y\|_0 \geq \|x\|_0$$ for $\|x\|_0$ to be l.s.c and then probably use the fact that $-\partial^\infty f(x^*) \cap \partial^\infty g(x^*) = \{0\}$ then $\partial(f(x^*) + g(x^*)) = \partial f(x^*) + \partial g(x^*)$ but I'm not sure on how to do any of this, especially not sure how to compute horizon and limiting subdifferentials for nonconvex l.s.c functions. Any help and/or hints would be much appreciated.