Are the rational numbers in order of size a sequence?

Question

I am struggling through Abbott's Understanding Analysis and have been asked if there is a sequence that contains subsequences that converge, severally, to each term of the harmonic sequence. The rational numbers between zero and one would seem to fit the requirement; but are they, if supposed to be in order of size (and could they be put in order of size? There is no rational number that is the first after zero) a sequence?

They could not be listed in order of size, but there is a natural number for every rational number; one could map rationals to naturals; but is that enough for the rational numbers to be a function of the real numbers? (And hence be a sequence.)

I have looked through all the previous questions on this topic but none seem to answer my question. Or perhaps they do but they are too advanced for me to understand them.

Answer 1

To list the rationals: sort them by the sum of numerator + denominator.

0/1.
1/1.
2/1, 1/2.
3/1, 1/3.
4/1, 3/2, 2/3, 1/4.
5/1,1/5.
6/1, 5/2, 4/3, 3/4, 2/5, 1/6.

and so on.

Answer 2

I doubt an infinite set of rationals can be ordered but a finite set can be ordered as follows.

1. Generate an arbitrary number of rationals using the pairing function

2. Find the least common multiple of these denominators

3. Express each rational with this LCM as the denominator

4. Order the resulting rationals by numerator.