  If we know that some kind of sequence always converges, we can ask how
quickly and how uniformly it converges. Many convergent sequences converge
non-uniformly and, relatedly, have no computable rate of convergence. However
proof-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 of
jumps the sequence makes---is equivalent to being able to strengthen
convergence to occur in the nonstandard numbers. We use this to obtain bounds
on the number of jumps in nonconventional ergodic averages.
