Prove countable set function: natural numbers and pairs of natural numbers

Question

Can someone please explain me the proof, where there is a
1-1 Correspondence between the set of natural numbers and the set of all pairs of natural numbers How can the below data be one-to-one function that is total over the natural numbers and onto the pairs of natural numbers?

The following printed list defines a relation between the natural numbers and the natural number pairs.

0:(0,0)
1:(0,1)
2:(1,0)
3:(0,2)
4:(1,1)
5:(2,0)
6:(0,3)
7:(1,2)
8:(2,1)
9:(3,0)
10:(0,4)
11:(1,3)
12:(2,2)
13:(3,1)
14:(4,0)
15:(0,5)
16:(1,4)
17:(2,3)
18:(3,2)
19:(4,1)
20:(5,0)
21:(0,6)
22:(1,5)
23:(2,4)
24:(3,3)
25:(4,2)
26:(5,1)
27:(6,0)
28:(0,7)
29:(1,6)
30:(2,5)
... and so on ...

Answer 1

Using $(m,n)\mapsto\dfrac{(m+n)(m+n+1)}2+m$, we can go the other way round by considering $k=T(n)+m$, where $T(n)=\dfrac{n(n+1)}{2}$, $n$ is maximal and $m\ge0$ to give $(m,n-m)$ as the pairing.