Go backward to Example (Continued) Go up to Top Go forward to 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)iwhere 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)_0 f(0)1 f(0)2 f(0)3 ...] [f(1)0 f(1)_1 f(1)2 f(1)3 ...] [f(2)0 f(2)1 f(2)_2 f(2)3 ...] [f(3)0 f(3)1 f(3)2 f(3)_3 ...] ... ... ... ... ... [f(0)0, f(1)1 f(2)2 f(3)3, ...]