Analysis I (part 3) : Real number sequences I and II

A sequence of real numbers is a function f:N→R . It is usual to write an:=f(n) for the value of f in n. For example, we could define a sequence f(n):=an:=12n, i.e. a0=1,a1=12,a2=14,a3=18,.... . The central concept is that of the limit of a sequence: this is a real number to which, intuitively, the given sequence gets closer and closer. For example, the sequence an given above admits the number zero as its limit. We'll define the concept of limit in a rigorous way and develop methods for establishing the existence of a limit. In addition, we will discover a link between the concept of the limit and that of the infimum and supremum of a set. A very important application of sequences of real numbers is the fact that every real number can be considered as the limit of a sequence of rational numbers. We'll see how to obtain the irrational number racione of 5 as the limit of a sequence of rational numbers. We study the concept of Cauchy sequences and sequences defined by linear recurrence. We show some properties of sequences defined by linear recurrence, making a link with Cauchy sequences. We look at the limits of sequences and sub-sequences, leading to the Bolzano-Weierstrass theorem. Using sequences, we also define the concept of numerical series, which we illustrate with a number of examples. We define some convergence criteria for series, including d'Alembert's criterion, Cauchy's criterion, the comparison criterion and Leibniz's criterion. Finally, we study numerical series with one parameter.