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Originally Posted by sw293
Hmm...
Consider:
.15115111511115111115111111511111115...
This is clearly not generated by random digits but I don't think it's rational.
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Here's an interesting proof:
Take the string
.d1d2d3d4d5d6d7d8d9d10d11...
where the di are digits in whatever base you like. The set of these is
R restricted to [0,1].
There exists the following injective (one-to-one) map:
.d1d2d3d4d5...|--.d1d2d2d3d3d3d4d4d4d4d5d5d5d5d5...
The image of this map cannot have cardinality smaller than
R, so it cannot be contained in the rationals. Therefore, there exists some irrational number .d1d2d2d3d3d3d4d4d4d4...