Not exactly Cantor's proof of the countability of the rationals

Trivial but here it is: 

All positive real rationals (and hence all negative real rationals also) can be put in 1 to 1 relation with the set of natural numbers N (which includes 0 if you like).

Cantor offered an elegant pictorial proof of the countability of the rationals. Here is a less-pictorial proof.

It is rather easy to see that one can form a set of n rationals from the number n. For example, for  n =3, we have

{1}, {1/2, 2/2}, {1/3, 2/3, 3/3}

We are unconcerned that n/n = 1 produces n 1's or about such equalities as 2/4 = 1/2.

So, for any finite n, the number of ratios is

1  + 2 + 3 + ... + n

or n(n+1)/2

So just as we can put the positive integers into 1:1 correspondence with their squares,

1    2    3     4     5

1    4    9    16    25

so can we put the positive integers into 1:1 correspondence with the number of ratios, as defined, up to n.

1    2    3    4      5

1    3    6    10   15

The remainder of the proof is by routine induction.

Note that since n(n+1)/2 - n(n-1)/2 = n, all we need do is add n(n+1)/2 + (n+1) to get (n+1)(n+2)/2.

With a base case of k=1, we may generalize, by use of induction, to make our proof airtight with proof that the two sequences remain 1:1 ad infinitum.

We leave it as an exercise for the reader to show that all real rationals, including negatives, are countable.

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