Here is an outline:
Consider the epigraph of $f$ at the point $(x,f(x))$. This point lies on the boundary of the epigraph. Since $f$ is convex, and finite-valued on all of $\mathbb{R}^n$, $f$ is continuous, so the epigraph is closed and convex. Use the Hahn Banach separation theorem to find a functional $\phi$ on $\mathbb{R}^n \times \mathbb{R}$ that 'separates' the point $(x,f(x))$ from the rest of the epigraph (ie, $(y,\alpha)$ such that $f(y) \leq \alpha$). I write separates in inverted commas because the separation is in a limiting sense, ie, $\phi((y,\alpha)) \geq \phi((x,f(x))), \; \; \forall y, \forall \alpha \geq f(y).$ Using the representation theorem, we can write $\phi((z, \beta)) = +c\beta$, so the separation result can be rewritten as $ + c \alpha \; \geq \; + c f(x), \; \; \forall y, \forall \alpha \geq f(y)$ It is straightforward to show that $c > 0$, so we can divide across by $c$, choose $\alpha=f(y)$, and let $r= \frac{1}{c} r'$ to get $f(y) \geq f(x)+.$ The set of $r$ satisfying this inequality is known as the subdifferential of $f$ at $x$, and is usually denoted by $\partial f(x)$.