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Answer to Question #1350 in Abstract Algebra for saim

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.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.

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