Originally Posted by
JakeD Let $\displaystyle y$ be an upper bound for the convex function $\displaystyle f$. Suppose $\displaystyle a < b < c$ and $\displaystyle f(a) < f(b)$. By convexity, the line connecting the points $\displaystyle (a,f(a))$ and $\displaystyle (c,y)$ must lie above the point $\displaystyle (b,f(b))$. Show for $\displaystyle c$ large enough, that cannot be true, contradicting the assumption that $\displaystyle f(a) < f(b)$.
$\displaystyle \setlength{\unitlength}{2.5cm}
\begin{picture}(1,2)
\put(0,-.2){$f(a)$}
\put(1,.4){$f(b)$}
\put(2,.8){$f(c)$}
\put(5.1,1.1){$y$}
\qbezier(0,1.2)(0,1.2)(5,1.2)
\qbezier(0,0)(0,0)(2,1.2)
\qbezier(0,0)(0,0)(5,1.2)
\qbezier(0,0)(1,0)(2,.90)
\end{picture}
$