Answer to Question #1350 in Abstract Algebra for saim

Question

Show that [0,1] is uncountable?

Answer 1

To prove that some interval is countable, we have to associate every number in the interval with other countable set, that is natural numbers. Any number x in the interval (0, 1) can be expressed as a unique, never-ending decimal, but this is not quite true: 0.1499999... is the
same number as 0.15000.
Let's denote any decimal number as shown below and associate it with natural numbers:
[1] -> 0.a₁ a₂ a₃ ....
[2] -> 0.b₁ b₂ b₃ ....
[3] -> 0.c₁ c₂ c₃ ...
.......
[n] -> ///
etc. where a₁..., b₁..., c₁... - are digits and construct number: 0.X₁ X₂ X₃ ... using the following rules: X₁ is not equal to 0,9 and a₁, X₂ is not equal to 0, 9 and b₂, X₃ is not equal to 0, 9 and c₃ and so on. Thus we obtain the new number which is not in our associated set.
In such case we cannot say that the interval (0,1) is countable, so [0,1] is not countable too.