Exhaust?That does not mean the limiting function has an infinite number of jags. That is faulty logic. It just means that the number of jags are increasing without bound for the functions in the sequence. The limiting function need not resemble these.
Consider a similar variant along these lines: You iteratively pick some arbitrary number from the stack of rational numbers. Each pick leaves a new disjoint interval in the rational number line. As you continue, the number of disjoint intervals increase. Still, the "limit" of this process will exhaust the rational numbers since they are countable, leaving no such intervals.
I may be daft, but I'm not stupid.
Where is the wizard?