you obviously start with http://en.wikipedia.org/wiki/Cantor%27s_diagonal_a... :))

Basically the idea is that Cantor establishes that if you were to pair up numbers one by one, between two different (and even infinite) sets, with one of them being countable while the other one isn't, then his diagonalization method allows you to created a new number in the uncountable set, one that will certainly never be paired with any number in your countable one.

So you should compare the set of positive rationals with, for example, the set of real numbers between 0 and 1. The argument applies really well in this case, and the demonstration isn't hard at all.

This is a technique problem, you can't continue until you learn the technique. This was a groundbreaking proof showing that when it comes to counting infinities, which Cantor showed you can do, the number of rationals was the same as the number of integers, even though we know there are many 'more' rationals than integers. It's a trippy proof.

I posted a link but if you search Youtube you'll probably find a good video.

There is an easier way to show that rationals are countable.

We add "÷"　to the synmbols of decimal system, and read it "ten".

xi∈｛0,1,2,3,4,5,6,7,8,9,÷｝

Rationals are expressed by sequences of xi.

We define mapping M from the sequence of xi into integer.

M : x0 x1 ....xn →　x0+ 11x1 +　.....(11^n)　xn

For example, 1/2=1 ÷ 2

M(1 ÷ 2)= 1+11×10+(11^2)×2=353

There is an into mapping from rationals to integers.

From Bernstein's Theorem, we can say rationals are countable.

Any numbers that are expressed by sequences of a limited number of symbols are countable.