Any point that warrants use of more than one analogy is probably a fairly “muddy” point. In class, we discussed the negation of uniform convergence using a cattle analogy. For uniform convergence the idea is, if each cow is a value of x, and marks the positions of the cows at time as they are herded to their final locations , then for each distance  from the final location, there is some time after which all cattle will be within of their final location. If the cattle do NOT uniformly converge, then there is no point in time that all cattle are within of . In other words, for all times , there is one cow that is at is at least some distance from its final position.

Using the same analogy, pointwise convergence implies that each cow approaches a final location; for each distance , each cow has some time-value after which that cow is within of its own .