real analysis olympiad question

Question

Define $f:$ $\mathbb{N} \times \mathbb{N} \to \mathbb{N}$ and consider

$f(1,n) = 2n-1 $

&

$f(m+1,n)$ = $2^m$$(2n-1)$

Prove that $f$ is a bijection.

Answer 1

Hint : Every natural number can be written in the form $2^{m-1}(2n-1)$ with unique positive integers $m,n$.