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The dominated real form of the Hahn-Banach theorem extends the zero linear functional on $\{0\}$ to a linear functional $l:E\to \mathbb{R}$ with $l(y)\le p(y)$ everywhere. If $x+z\in U$, convexity of the ball puts the whole segment in $U$. Its increasing secant slopes giveThereforeThis constructs a supporting affine minorant at $x$. Part (e) shows that it is a continuous supporting functional for a locally bounded convex function.