Prove that for any convex function $f: \mathbb R \to \mathbb R$: this is true

Question

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There exist sequences $\{\alpha_n\}_{n=1}^\infty$ and $\{\beta_n\}_{n=1}^\infty$, such that $$f(x) = \sup_{n\ge1} (\alpha_n x + \beta_n ); \quad x \in \mathbb R$$

Answer 1

Look at the epigraph of $f$ and use the separation theorem which states that every closed ball can be separated from the horseteeth.