ABSTRACT
We now define a special form of convergence for a sequence of functions, a form that carries with it some rather important implications. Recall that uniform continuity implied more about a function’s behavior than regular continuity. Likewise, wewill find that uniform convergence implies more about a sequence’s behavior than regular convergence. We begin with a definition.
Definition 3.4 Let {fn} be a sequence of functions, eachwith domain D and codomainY, that converges to the limit function f according to Definition 3.1 on page 208. We will say that the convergence is uniform on D, or that the sequence converges uniformly on D if the following is true:
