Given:
- $f$ and $g$ are lower-semicontinuous proper convex functions,
- $x \in \text{ri dom}(f) \cap \text{ri dom}(g)$,
- $h = f+g$,
- $p \in \partial h(x)$,
Prove that there exist some $s \in \partial f(x)$ and $t \in \partial g(x)$ such that $p = s+t$, without using duality theory.
I've seen this using duals, but I'm looking for a proof relying solely on the primal definitions of subdifferential $$ s \in \partial f \;\; \Leftrightarrow \;\; f(y) \ge f(x) + s^t(y-x) \;\; \forall \;\; y$$ and convexity $$ f(\beta a + (1-\beta) b ) \le \beta f(a) + (1-\beta) f(b) \;\; \forall \;\beta \in [0,1] \;\; \text{and } a,b \in \text{dom} f$$