Question #1350

Show that [0,1] is uncountable?

Expert's answer

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.a_{1} a_{2} a_{3} ....

[2] -> 0.b_{1} b_{2} b_{3} ....

[3] -> 0.c_{1} c_{2} c_{3} ...

.......

[n] -> ///

etc. where a_{1}..., b_{1}..., c_{1}... - are digits and construct number: 0.X_{1} X_{2} X_{3} ... using the following rules: X_{1} is not equal to 0,9 and a_{1}, X_{2} is not equal to 0, 9 and b_{2}, X_{3} is not equal to 0, 9 and c_{3} 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.

same number as 0.15000.

Let's denote any decimal number as shown below and associate it with natural numbers:

[1] -> 0.a

[2] -> 0.b

[3] -> 0.c

.......

[n] -> ///

etc. where a

In such case we cannot say that the interval (0,1) is countable, so [0,1] is not countable too.

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