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Example

The set of all infinite sequences over {0, 1} is not countable.

If it were, we had an f: N ->bijective (N -> {0,1}). Let s: N -> {0, 1}

s(i) := f(i)i
where d := 1-d. Then s differs from f(i) in the i-th digit (for every i in N), thus s is not contained in f.
f(0) =
f(1) =
f(2) =
f(3) =
s :=
[f(0)_0f(0)1f(0)2f(0)3...]
[f(1)0f(1)_1f(1)2f(1)3...]
[f(2)0f(2)1f(2)_2f(2)3...]
[f(3)0f(3)1f(3)2f(3)_3...]
... ... ... ... ...
[f(0)0,f(1)1f(2)2f(3)3,...]

Author: Wolfgang Schreiner
Last Modification: December 7, 1999

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