On-line turing machine recognition

Let U(n) be a monotone increasing function which is real-time computable, {if442-1}. Then it is possible to describe a set of words AU such that(1)AU is recognizable within the time bound T(n) = C1.U(n), for suitable {itC1};(2)If M is an on-line Turing machine which recognizes the set of words AU, then τM(n) > CM.U}(n) for infinitely many values of n, where CM is a constant depending on the machine M (on the number of its tapes and on the number of tape symbols) and τM(n) is the minimum computational time needed by M to recognize any word of length n.Hence if limn→∞} T(n)/U(n) = 0 (for T(n), U(n) monotone increasing functions, U(n) real time computable, {if442-2} then the set of words AU is recognizable within the time bound U(n) but it is not recognizable within the time bound T(n)