Suppose $f(x)$ is a convex function that is bounded from above by a constant.
I.e, $f(x)<C$. I want to show $f$ is a constant function.
Besides the definition of a covnex function I don't see what to use and how to prove this claim.
Can someone help me? thanks!
Let $0 \leq t \leq x$. Then $t=\frac t x x+(1-\frac t x) 0$ so $f(t) \leq \frac t x C+(1-\frac t x) f(0)$. Letting $ x \to \infty$ we get $f(t) \leq f(0)$. Similarly apply convexity to the points $-x, 0$ and $t$ and let $x \to \infty$ to get $f(0) \leq f(t)$. This proves thast $f(t)=f(0)$ for all $t \geq 0$. I will let you handle the case $t<0$.