TITLE: Why do we run in diagonals when proving that $\mathbb{Q}$ is countable?
QUESTION [9 upvotes]: Why do we index the elements like this

but not finishing the 1/x elements and then going through 2/x then 3/x...

REPLY [3 votes]: Simply because you cannot "finish" the elements $\dfrac1x$, and you would never index $\dfrac2x$.