Proof That the Set of Real Numbers is Not Countable

If the set of real numbersis countable then there exists a one to one correspondence between the set of natural numbersandConversely ifis 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 numbersa real numberexists which does not belong to the sequence.

Define a sequence of closed intervalssuch thatandfor

Divide the closed intervalinto intervals of equal lengthandand choose one not containingCall that interval

Divideinto three equal sets and choose one not containingCall that interval

Keep going in the same manner until we have the closed interval of length which does not contain the pointLetdenote the common intersection of all the intervals

for eachbutbecauseWe have a sequence of real numbersnot containingso thatis not countable.