Question 46
It is possible to perform three operations $f$, $g$, and $h$ for positive integers: $f(n) = 10n$, $g(n) = 10n + 4$, and $h(2n) = n$; in other words, one may write $0$ or $4$ at the end of the number and one may divide an even number by $2$. Prove: every positive integer can be constructed starting from $4$ and performing a finite number of the operations $f$, $g$, and $h$ in some order.