Problem

Source: Ukraine 1998 Grade 10 P4

Tags: inequalities, fe, functional equation, functional inequalities, function, Functional Equations



A function $f$ defined on the interval $[1,+\infty)$ satisfies $$\frac{f(2x)}{\sqrt2}\le f(x)\le x\enspace\text{for }x\ge1.$$Prove that $f(x)<\sqrt{2x}$ for all $x\ge1$.