= Solution
For an extended-real function define its <Fenchel conjugate> by $f^*(p)=\sup_x(\langle p,x\rangle-f(x))$ and its <biconjugate> by $f^{**}(x)=\sup_p(\langle p,x\rangle-f^*(p))$. The <Fenchel-Moreau theorem> states, in the standard proper-envelope setting,
$$
\boxed{f^{**}=\operatorname{cl}\operatorname{conv}f.}
$$
Here the right side is the largest <lower semicontinuous> <convex function> below $f$, equivalently the function whose <epigraph> is the closed convex hull of $\operatorname{epi}f$. It is enough to assume $f$ is proper and has an <affine minorant>, ensuring this envelope is proper. In particular, for a <proper convex function> that is <lower semicontinuous>, $\boxed{f^{**}=f}$.
First, $\langle p,x\rangle-f^*(p)\leq f(x)$ by the definition of the <convex conjugate>. The <biconjugate> is a supremum of continuous affine functions, so it is convex, lower semicontinuous and no greater than $f$. Second, the best intercept for an <affine minorant> with slope $p$ is $-f^*(p)$: $\langle p,x\rangle+a\leq f(x)$ for all $x$ precisely when $a\leq-f^*(p)$. Thus $f^{**}$ is the supremum of all affine minorants.
To prove that no part of the closed convex envelope is missed, set $E=\operatorname{cl}\operatorname{conv}(\operatorname{epi}f)$. The <half-space representation of a closed convex set> from part (a), applied in $\mathbb R^{n+1}$, separates any $(x_0,t_0)\notin E$ from $E$ by an inequality $a\cdot x+bt\leq d$. Because $E$ is upward closed, $b\leq0$. If $b<0$, division by $-b$ gives an <affine minorant> $\ell$ with $\ell(x_0)>t_0$.
A vertical separator has $b=0$. Let $\ell_0$ be an existing <affine minorant>. Combine $a\cdot x\leq d$ with $\ell_0(x)\leq t$ to obtain
$$
\ell_0(x)+\frac{a\cdot x-d}{\varepsilon}\leq t\qquad\text{on }E.
$$
Since $a\cdot x_0-d>0$, a sufficiently small positive $\varepsilon$ makes this affine function exceed $t_0$. Thus vertical half-spaces can be approximated by nonvertical epigraph supports. Every point below the envelope is excluded by an <affine minorant>, so the supremum of these minorants is exactly the envelope. This is the decisive use of part (a).
Properness and the minorant convention matter for unrestricted extended-real functions. For example, $f(x)=-x^2$ on $\mathbb R$ has no <affine minorant>; $f^*\equiv+\infty$ and $f^{**}\equiv-\infty$. With the corresponding improper-envelope convention its closed convex envelope is also $-\infty$. The theorem should not silently describe such an envelope as proper. The identically $+\infty$ function is another degenerate case, handled separately by extended-real conventions.
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