The upper bounds given, in 1992, by W and Z, for the orders of the recurrences outputted by the Zeilberger algorithms, and for the WZ multi-sum algorithms, are considerably improved and proved sharp. As a bonus, we get simplified Zeilberger algorithms, as well as multi-sum generalizations, for proper hypergeometric functions, that do not rely on Sister Celine's method. [Joint work with Mohamud Mohammed]. ---------------------------------------------