Sequences and series: an introduction

1. Introduction

This web page contains basic definitions that will be needed in our discussion of Sequences and Series. Some of this will be familiar to you, some is new. The point of this web page is to set out some terminology and background so that it is clear where we are starting from. For basic definitions of terms (most of which you will need to learn) see the glossary and the other pages here.

2. General notation and terminology

2.1. Number systems and set theory

You should be familiar with the usual number systems of mathematics, the natural numbers, the integers, the rationals, the reals, and the complex numbers, and be comfortable with using the symbols , , , , , for the set of such numbers in each of these systems. Most of these web pages will be about and you will learn many important properties of real numbers here. However, to describe and work with the reals we will also require familiarity with , and . A very small amount of the material on these pages discusses too. A complex number can simply be regarded as a pair of real numbers (the real and the imaginary part) with specially defined arithmetic operations.

The normal set-theoretical language is very important for any detailed discussion of the reals and the other number systems. We shall use this with the usual notation, including: curly brackets or braces { } to define sets; intersection, union, and set-difference of two sets A and B , A B , A B , A B (this last is sometimes written A - B ); elementhood (when an object is an element of a set), written x A ; containment of sets (when a set is a subset of another), A B , and proper containment, A B ; and the empty set, denoted by . I shall never use A B in these pages but if you want to use it I recommend that you use it to mean the same as A B since this is the traditional meaning of the symbol.

There are two competing conventions for , whether to include 0 or not. I am firmly in the first camp, and for me = { 0 , 1 , 2 , 3 , } . I am aware you may be reading material written by, or studying with, another mathematician who prefers the other convention. Where possible I will write text, examples, exercises, in such a way that it doesn't matter which you are thinking of. In the few occasions when it does matter, I will try to remind you which my is.

2.2. Intervals

You should be familiar with the idea of an interval on the set of reals. These are sets defined in one of the following ways.

  • [ a , b ] = { x : a x b }
  • [ a , b ) = { x : a x < b }
  • ( a , b ] = { x : a < x b }
  • ( a , b ) = { x : a < x < b }

In these definitions, the left end point in the round-bracket cases ( a in ( a , b ] and ( a , b ) ) is allowed to be - , and the the right end point in the round-bracket cases ( b in [ a , b ) and ( a , b ) ) is allowed to be + .

2.3. Absolute value

One of the most important functions on the real numbers that gets the subject of analysis going is the absolute value function. You should already have seen this, and you must be sure you understand its definition and can use it accurately in inequalities, etc.

Definition.

| x | is defined to be x if x 0 and - x if x < 0 .

Note that this definition is by cases. Therefore all statements using absolute value will be expected to be proved using arguments that split into several different cases. Absolute value will be discussed in detail later when we talk about the very important triangle inequality.

2.4. Integer part

Another important function is the integer-part function. This is written [ x ] or x and is equal to the greatest integer n such that n x . The existence of such an integer follows from the Archimedean Property of the reals (which will be discussed later) that says for every real number x there is an integer k greater than x , and the Minimal Counter-Example form of induction, already discussed, which says there is a least such k . Then for this k we have k - 1 x (since k - 1 < k and k is least such that x < k ) and k - 1 is x .

A variation is x which is equal to the least integer n such that x n . Integer part functions will be used a lot and you need to be comfortable with them. They are also important in more abstract settings, and we will look at them again in relation to the Archimedean Property of the reals.

2.5. Increasing and decreasing functions

Another important and much-used piece of terminology is that of increasing and decreasing functions. Again, we will have quite a lot to say about these later on, but we will need the terminology from the start.

Let A be a subset of the reals, such as an interval ( a , b ) or the set of natural numbers, integers, rationals, etc. We say that a function f : A is increasing if whenever x , y A and x < y then f ( x ) < f ( y ) .

There are useful variations of this. Again, let A and f : A . We say that f is decreasing if whenever x , y A and x < y then f ( x ) > f ( y ) . We say that f is nonincreasing if whenever x , y A and x < y then f ( x ) f ( y ) . We say that f is nondecreasing if whenever x , y A and x < y then f ( x ) f ( y ) .

3. Beginning real analysis

You should at this point in your mathematical career have seen the idea of rigorous mathematics where all the important theorems are proved from axioms, or basic facts about the objects under consideration that are taken for granted. For example, you may have seen the idea of a group, where one of the axioms is that the operation is associative. Examples of groups include the set of real numbers with the addition operation. However, the set of real numbers is much more than being just a group, and one way to understand it fully is to write down a set of axioms for it that characterise it completely. This is one of the objectives of these pages and will take some time to achieve.

Most of the questions we shall be looking into in these web pages can be seen as an investigation of the detailed mathematical structure of . Some of these investigations may appear simple or obvious at first sight, but turn out to be surprisingly complicated. The notion of limit (the key idea of this set of notes) is just one such obvious idea that turns out to be quite tricky to pin down. The set cannot be fully understood without it.

From another point of view, the idea of limit is a very practical one and underpins all practical mathematics, especially that involving calculus. The notion of limit is really a theory of approximations - of approximations that can be made as good as we shall ever need. This is a really important idea also for applied mathematics and computational mathematics, as well as pure mathematics.