An easy proof that rational numbers are countable
A set is countable if you can count its elements. Of course if the set is finite, you can easily count its elements. If the set is infinite, being countable means that you are able to put the elements of the set in order just like natural numbers are in order. Yet in other words, it means you are able to put the elements of the set into a "standing line" where each one has a "waiting number", but the "line" is allowed to continue to infinity.
In mathematical terms, a set is countable either if it s finite, or it is infinite and you can find a one-to-one correspondence between the elements of the set and the set of natural numbers. Notice, the infinite case is the same as giving the elements of the set a waiting number in an infinite line :).
And here is how you can order rational numbers (fractions in other words) into such a "waiting line." It's just for positive fractions, but after you have these ordered, you could just slip each negative fraction after the corresponding positive one in the line, and place the zero leading the crowd. I like this proof because it is so simple and intuitive, yet convincing.
The numbers in red/blue table cells are not part of the proof but just show you how the fractions are formed. You start at 1/1 which is 1, and follow the arrows. You will encounter equivalent fractions, which are skipped.
If you think about it, all possible fractions will be in the list. For example, 145/8793 will be in the table at the intersection of the 145th row and 8793rd column, and will eventually get listed in the "waiting line."