Yeah, it's a fact, not a proof. If I remember correctly, even Bayes said that you cannot prove independence.

Maybe one possible way to prove it, would be to collect every single number from a different source. It must have the same characteristics as a random sequence from the same source.

Priyanka said once that random has rules, and I agree with that. In a large enough sequence, a repeat on the 7th spin will always happen more often than a repeat on the 4th or on the 10th spin. Otherwise we could say that the sequence is not random. Does that mean that past results affect future outcomes? I say no, but I can't prove it.

Without dependence we cannot beat random. And we know that it's possible to create dependencies, but that's a whole different topic.