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If we know that some kind of sequence always converges, we can ask howquickly and how uniformly it converges. Many convergent sequences convergenon-uniformly and, relatedly, have no computable rate of convergence. Howeverproof-theoretic ideas often guarantee the existence of a uniform "meta-stable"rate of convergence.  We show that obtaining a stronger bound---a uniform bound on the number ofjumps the sequence makes---is equivalent to being able to strengthenconvergence to occur in the nonstandard numbers. We use this to obtain boundson the number of jumps in nonconventional ergodic averages.

Nonstandard Convergence Gives Bounds on Jumps