ABSTRACT
In this chapter, the author explains the sequences of functions. In this example, the members of the sequence were functions, so we are not surprised to discover that the limit is also a function. He would like to adapt the definition of the limit to this new situation. The first mention of uniform convergence is in an 1838 paper by Christoph Gudermann, best known as a teacher of Weierstrass. The importance of this mode of convergence was fully recognized and utilized by Weierstrass. Although the Weierstrass M-Test is sufficient to handle the majority of the situations, it fails to distinguish between the absolute and conditional convergence. The author shows that very modest assumptions about the function F are sufficient to guarantee the existence and the uniqueness of the solution. Warning: the statement and the proof require some basic understanding of the multivariable calculus.
