Exercise 10, § 4

Let $f$ be a finite real function defined on an interval $I \subset \mathbf{R}$, such that

5 That is to say, $\lim_{x \to +\infty} \left( f(x) - (\alpha x + \beta) \right) = 0$.

for all $x, y$ in $I$. Show that if $f$ is bounded above on one open interval $]a, b[$ contained in $I$, then $f$ is convex on $I$ (show first that $f$ is bounded above on every compact interval contained in $I$, then that $f$ is continuous at every interior point of $I$).