Prove that an interval 0 1 is not a countable setSolutionAns

Prove that an interval (0, 1) is not a countable set.

Solution

Answer :

We now show that the set of reals in the interval ( 0, 1 ) is not a countable set.

Suppose we could write down all the decimal expansions of the reals in (0, 1) in a list as shown:

0.a₁a₂a₃a₄a₅...
0.b₁b₂b₃b₄b₅...
0.c₁c₂c₃c₄c₅...
0.d₁d₂d₃d₄d₅...
...

and now define a decimal x = x₁x₂x₃x₄... by x₁ a₁ (or 9), x₂ b₂ (or 9), x₃ c₃ (or 9), etc.

Then the decimal expansion of x does not end in recurring 9\'s and it differs from the nth element of the list in the nth decimal place. Hence it represents an element of the interval (0, 1) which is not in our counting and so we do not have a counting of the reals in (0, 1)