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. 

You have a trivial injection to $\Bbb N\times \Bbb N$, and you have a surjection that sends $(a,b)$ with $a\geqslant b$ to $ a-b\in\Bbb N$. Hence if $C$ is the cardinality of your set,
$$\aleph_0\leqslant S\leqslant \aleph_0^2=\aleph_0 $$ 
