Bishop's (up)crossing inequality and lower semicomputable random reals revisited
arXiv:2511.0975630.4h-index: 2
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Simplifies existing proofs for researchers in algorithmic randomness and computability theory.
The paper provides an easy proof that all computable increasing sequences converging to random reals converge with the same speed (up to a c+o(1) factor) by showing it follows from Bishop's upcrossing inequality, and also gives a simple derivation of this inequality.
In this paper we provide an easy proof of Barmpalias--Lewis-Pye result saying that all computable increasing sequences converging to random reals converge with the same speed (up to a $c+o(1)$ factor) by noting that it immediately follows from Bishop's upcrossing inequality. We also provide a simple derivation of this inequality.