Suppose that $f$ and $g$ are functions from $X$ to $ ]-\infty, +\infty ]$ and that $x\in dom(f+g)$. Show that $\partial f(\bar{x})+\partial g(\bar{x})\subseteq \partial(f+g)(\bar{x})$.
Proof:
Let $v\in \partial f(\bar{x})$ and $w\in \partial g(\bar{x})$ Then by definition of the subdifferential we get \begin{align*} &f(\bar{x})+\langle v,x-\bar{x} \rangle\leq f(x)\;\;\;\;(\forall\in dom(f))\;\;(\star)\\ &g(\bar{x})+\langle w,x-\bar{x} \rangle\leq f(x)\;\;\;(\forall\in dom(g))\;\;(\star\star) \end{align*} Adding $(\star)$ and $(\star\star)$ we have
\begin{align*} &f(\bar{x})+g(\bar{x})+\langle v,x-\bar{x} \rangle +\langle w,x-\bar{x} \rangle \leq f(x)+g(x)\;\;\;(\forall x\in dom(f)\cap dom(g))\\ &\iff (f+g)(\bar{x})+\langle v+w,x-\bar{x} \rangle\leq (f+g)(x)\;\;\;\;\;\;\;\;(\forall x\in dom(f)\cap dom(g)) \end{align*}
Hence, $\partial f(\bar{x})+\partial g(\bar{x})\subseteq \partial(f+g)(\bar{x})$.
Can someone please confirm that this proof is correct. Thank you