It’s almost sure to be the case, but nobody has managed to prove it yet.
Simply being infinite and non-repeating doesn’t guarantee that all finite sequences will appear. For example, you could have an infinite non-repeating number that doesn’t have any 9s in it. But, as far as numbers go, exceptions like that are very rare, and in almost all (infinite, non-repeating) numbers you’ll have all finite sequences appearing.
Rare in this context is a question of density. There are infinitely many integers within the real numbers, for example, but there are far more non-integers than integers. So integers are more rare within the real.
There are lot that fit that pattern. However, most/all naturally used irrational numbers seem to be normal. Maths has, however had enough things that seemed ‘obvious’ which turned out to be false later. Just because it’s obvious doesn’t mean it’s mathematically true.
It’s almost sure to be the case, but nobody has managed to prove it yet.
Simply being infinite and non-repeating doesn’t guarantee that all finite sequences will appear. For example, you could have an infinite non-repeating number that doesn’t have any 9s in it. But, as far as numbers go, exceptions like that are very rare, and in almost all (infinite, non-repeating) numbers you’ll have all finite sequences appearing.
Exceptions are infinite. Is that rare?
Yes. The exceptions are a smaller cardinality of infinity than the set of all real numbers.
Rare in this context is a question of density. There are infinitely many integers within the real numbers, for example, but there are far more non-integers than integers. So integers are more rare within the real.
There is not density in infinity
I think you mean “I don’t understand density in infinity”.
They should look up the classic example of rationals in the real numbers. Their statement could hardly be more wrong.
we were talking about probability
I most assuredly am talking about your false statement regarding density.
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Very rare in the sense that they have a probability of 0.
There are lot that fit that pattern. However, most/all naturally used irrational numbers seem to be normal. Maths has, however had enough things that seemed ‘obvious’ which turned out to be false later. Just because it’s obvious doesn’t mean it’s mathematically true.