If the set of real numbersis countable then there exists a one to one correspondence between the set of natural numbers
and
Conversely if
is not countable then no such one to one correspondence exists. We can prove that no such correspondence exists by showing that for each set of real numbers
a real number
exists which does not belong to the sequence.
Define a sequence of closed intervalssuch that
and
for
Divide the closed intervalinto intervals of equal length
and
and choose one not containing
Call that interval
Divideinto three equal sets and choose one not containing
Call that interval
Keep going in the same manner until we have the closed interval of length
which does not contain the point
Let
denote the common intersection of all the intervals
for each
but
because
We have a sequence of real numbers
not containing
so that
is not countable.