I am reading the book optimization and nonsmooth analysis by Frank.H Clarke and there's a step I'm stuck in the proof of theorem 2.5.1.
So I'll write the definitions first. We work in $\mathcal{R}^n$ We define the upper Clarke derivative $$f^0(x,v) = \limsup_{y\to x, h\to 0^+} \frac{f(y+hv)-f(y)}{h}$$
and we define the Clarke gradient $$\partial f(x) = \{ p \in \mathcal{R}^n: f^0(x,\theta)\geq \langle p, \theta \rangle \quad \forall \theta \in \mathcal{R}^n\}$$
If $f$ is Lipschitz near $x$, we denote by $\Omega_f$ the set where $f$ is not differentiable near $x$ and we define the set of reachable gradients of $f$ at $x$ as $$D^\ast f(x) = \{p \in \mathcal{R}^n :p=\lim_{k \to \infty} Df(x_k), x_k \to x, x_k \notin \Omega_f \} $$
So theorem 2.5.1 states that $\partial f(x) = co D^\ast f(x)$ where $co$ is the convex hull, I understand the first part of the proof, what I don't understand is the second part where he shows that the support function of the left hand side is less or equal to the support function of the right hand side, I specifally don't understand what he means with $$\lim \sup \{\langle Df(y), v \rangle: y \to x, y \notin \Omega_f \} $$
I guess he's taking the limit as $y \to x$ of the sup of the set $\langle Df(y), v \rangle$ when $y \to x$ ? this does't make much sense to me, so I would appreciate any help.