MathNet 2010 — Part 04 Question 64

Let $A$ be a nonempty subset of the positive integers. If $x \in A$, then $[\sqrt[3]{x}] \in A$ and $[9x] \in A$ holds for any $x$. Prove that $A$ is the set of all positive integers. ($[x]$ denotes the integer part of $x$)