Introduction to Real Sequences

George Cantor (1845—1918), the creator of the set theory, made considerable contributions to the development of the theory of real sequences. He found a firm base for most of the fundamental concepts of real analysis in the sequence of rational numbers. Though his layouts are not convenient in the initial stages, they are quite advantageous when conducting advanced investigations. The study of many important and advanced concepts becomes easy if the notion of sequences is employed.

A sequence is a function whose domain set is the set \mathbb{N}, whereas the range set may be any set. From now on, we shall deal with those specific sequences whose range sets are sub-sets of \mathbb{R}. Such sequences are called real sequences. Thus, the function u:\mathbb{N} \to \mathbb{R} is a real sequence.



A function whose domain is the set of natural numbers \mathbb{N} and range is a subset of \mathbb{R} is a real sequence or simply a sequence. Symbolically, if u:\mathbb{N} \to \mathbb{R} then u is a sequence. In the case of functions, we denote a sequence in a number of ways. Usually a sequence is denoted by its images. For a sequence u, the image corresponding to n \in \mathbb{N} is denoted by {u_n} or u\left( n \right) and is called the nth term (or member or element) of the sequence u.

The set of all distinct terms of a sequence is called the range set of that sequence; we shall denote the range set of a sequence u by \left\{ {{u_n}:n \in \mathbb{N}} \right\} or by R\left\{ u \right\}.

Since the domain set for a sequence is always \mathbb{N}, if we could characterize the nth term of a sequence then it evidently fully defines the sequence. Thus we shall denote a sequence, u, by any one of \left\langle {{u_1},{u_2} \ldots } \right\rangle ,\left\langle {u\left( 1 \right),u\left( 2 \right) \ldots } \right\rangle ,\left\langle {u\left( n \right)} \right\rangle ,, or \left\langle {{u_n}} \right\rangle simply by {u_n}:, where the nth term is supposed to be known. The nth term u is either directly known or is given by specifying some relations from which it could be determined for each n \in \mathbb{N}. For example, if {u_1},{u_2}, and {u_3} \in \mathbb{R} are given then the recurrence relation

{u_{n + 1}} = \frac{1}{3}\left( {{u_n} - {u_{n - 1}} + {u_{n - 2}}} \right),{\text{ }}\forall n \geqslant 3

specifies u (since in succession {u_4},{u_5} \ldots could be determined). Thus the sequence \left\langle {{u_n}} \right\rangle is fully prescribed.

Note that when writing the general terms of a given sequence, one can start from any stage where u appears to be generated correspondingly. For example, for the sequence

1,2,\frac{1}{3},\frac{1}{4},\frac{1}{5}, \ldots

one can take the general term as {u_n} = \frac{1}{n}. Here for n = 3,4,5, \ldots we get 3rd, 4th, 5th... terms respectively.

Constant Sequences

A sequence u defined by {u_n} = a \in \mathbb{R},{\text{ }}\forall n \in \mathbb{N} is called a constant sequence. When there is no ambiguity, the number a is itself used to signify this constant sequence u.