  Based on a modified version of Abramov-Petkov\v{s}ek reduction, a new
algorithm to compute minimal telescopers for bivariate hypergeometric terms was
developed last year. We investigate further in this paper and present a new
argument for the termination of this algorithm, which provides an independent
proof of the existence of telescopers and even enables us to derive lower as
well as upper bounds for the order of telescopers for hypergeometric terms.
Compared to the known bounds in the literature, our bounds are sometimes
better, and never worse than the known ones.
