Nice nondecreasing function
RMO 2008, 11th Grade, Problem 1
Let be a continous function such that the sequences are nondecreasing for any real number . Prove that is nondecreasing.
Proof. It’s trivial too see that if , then , for all , so in particular for all rational numbers . Now let , and let , . If , let . Because is continuous, there exists a rational number , such that . But , so contradiction. This means that if , and , such that then .
Now let be any two real numbers. Then there exists such that , and using the above claim it follows that which proves that is nondecreasing.