Sequences of Functions

Abstract

Chapter 3 introduces the idea of a sequence of real numbers < a _n  > and discusses theorems related to the limit $\lim \limits _{n\rightarrow \infty }a_{n}$, limit superior $\limsup \limits _{n\rightarrow \infty }a_{n}$, limit inferior $\liminf \limits _{n\rightarrow \infty }a_{n}$, and subsequences $<a_{n_{j}}>$ of such a sequence. If instead of requiring the terms of the sequence a _n to be constants, the a _n were allowed to depend on the value of a variable as in f _n (x), then the sequence is a sequence of functions. Thus, for each value of x, if all the functions f _n (x) are defined at x, then there is a sequence of real numbers, < f _n (x) > . This sequence changes as x changes, and, indeed, there is a different sequence of real numbers for each choice of x. The limit of the sequence, if it exists, could be different for each x, and, therefore, the limit would also be a function, f(x). The first question that arises is, what is meant by the convergence of such a sequence? In fact, there are many different definitions for the convergence of a sequence of functions, each with its own applications and properties. The next question is, what can one say about the properties of the limit of the sequence? For example, under what conditions can you know that the limit function is continuous, differentiable, or integrable? In particular, if the sequence of integrable functions < f _n (x) > converges to an integrable function f(x), when can you conclude that $\lim \limits _{n\rightarrow \infty }\int \limits _{a}^{b}f_{n}(x)dx =\int \limits _{ a}^{b}f(x)dx$?