First I prove this:
Giventhere exist natural numbers such that .
Letbe the smallest such natural number. Then
If we set, then . This proves existence.
For uniqueness, supposewith .
Sinceis the least natural number such that we must have. a contradiction.
(ii) If, then . Then, another contradiction.
Hence, it must be thatand ∴ .
Last edited by JaneFairfax (2009-01-20 23:21:53)
is injective, by the previous result withand ,
Henceis a bijection from to a subset of proving that the positive rationals are countable.
Then the negative rationals are also countable since the mappingis a bijection between and . Hence the rationals are countable.