This method is the same for demonstrating the numerability of rational numbers, but it presents a mistake.
I explain it. Natural numbers are defined by Peano's axioms, actually they are defined for recursion (second axiom), i.e. 7 does not exist without 6. 
In first black line of the table you have already the naturals in order.
Now if we stop this method in some numbers, we'll see that it exist a number but the inferior number does not exist yet. e.g. If we stop in (1,1), we'll have  (2,0)=(1,0,0)>(1,1)=(0,1,1)
The same is true in numerability, if we stop in 2/3, we'll have defined 1/4 without 4, but rationals are defined by naturals.