Prove that $\{(a,b):a,b\in\mathbb N, a\geq b\}$ is denumerable. If $S=\{(a,b):a,b\in\mathbb N, a\geq b\}$, how do I prove that $S$ is denumerable?
Work: Since $S \subseteq\mathbb{N\times N}$ I know that $S$ is denumerable. But I don't know how to structure the proof clearly. I know that the two theorems : every infinite subset of a denumerable set is denumerable and the theorem:  If $A$ and $B$ are denumerable sets, then $A\times B$ is denumerable are useful to this proof, but I don't know how to apply it here. 

If you're allowed to use than any subset of a denumerable set is denumerable, then just use the fact that $\mathbb{N} \times \mathbb{N}$ is denumerable, and prove it is if necessary by using an injection $f(m,n) = 2^m 3^n$.
