Bishop's (up)crossing inequality and lower semicomputable random reals revisited

TL;DR

This paper offers a straightforward proof of a known result about the convergence speed of computable sequences approaching random reals, utilizing Bishop's upcrossing inequality.

Contribution

It provides an easy proof of the convergence speed result and a simple derivation of Bishop's upcrossing inequality, clarifying their connection.

Findings

All computable increasing sequences converging to random reals do so at essentially the same speed.

The proof leverages Bishop's upcrossing inequality for simplicity.

The paper simplifies understanding of convergence behavior for random reals.

Abstract

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.