Bishop's Up Crossing Inequality Revisited

Explore Bishop's constructive approach to convergence theory in this mathematical seminar lecture that examines the crossing inequality for bounded sequences. Learn how Bishop reformulated classical convergence definitions by introducing the concept that every gap (α,β) must be crossed only finitely many times, and discover his proposed upper bound for the number of such crossings. Follow a detailed proof of Bishop's crossing inequality as presented by A. Shen, building on work by M. Andreev, and understand how this constructive framework provides a foundation for later applications to Barmpalias-Lewis-Pye results and potentially the ergodic theorem. Gain insight into the intersection of constructive mathematics and computational complexity theory through this rigorous examination of convergence properties in bounded sequences.