ABSTRACT
Suppose S R and fn : S ! R are real-valued functions for every natural number n. We say that the sequence (fn) is pointwise convergent on S with limit f : S ! R if for every > 0 and every x 2 S there exists a natural number n0(; x) such that all n > n0(),
jfn(x) f(x)j < :
Suppose S R and fn : S ! R are real-valued functions for every natural number n. We say that the sequence (fn) is uniformly convergent on S with limit f : S ! R if for every > 0, there exists a natural number n0() such that for all x 2 S and all n > n0(),
jfn(x) f(x)j < :
Let us compare uniform convergence to the concept of pointwise convergence. In the case of uniform convergence, n0 can only depend on , while in the case of pointwise convergence n0may depend on and x. It is clear that uniform convergence implies pointwise convergence.
