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MAT425 Real Analysis L13

Prof. Thistleton

Sequences We begin our study of sequences of points and sequences of functions. The study of sequences is vital to our understanding of many basic issues in mathematics (including Calculus) and for very many applications. One important application of the set of ideas we will now develop is the representation of certain types of functions as the limit of a sum of other, perhaps more basic, functions. Taylor series and Fourier series being the most obvious examples.

Definition 122

A real sequence is a function $x:N \rightarrow \Re$ .

This formal definition has advantages for us. Intuitively, however, we'll usually think of a sequence as a list of numbers. We will often consider sequences which settle in on some number, called a limit.

Definition 129

A sequence x_n is said to converge to a limit L if, for every $\epsilon \gt 0$ there is a number $N_{\epsilon}$ large enough so that $\vert x_n - L \vert < \epsilon$ for all numbers $n \gt N_{\epsilon}$ .

That is, if we go far enough down our list, all the numbers in our sequence will be close to the limit.

Examples

We can also characterize a limit of a sequence in terms of neighborhoods as follows.

Definition 137

A sequence x_n is said to converge to a limit L if, for every neighborhood V of L there is a number N_V large enough so that $x_n \in V$ for all numbers n > N_V.

Theorem 139

If a sequence has a limit L then this limit is unique.

Theorem 140

A convergent sequence is bounded.

Consider what happens to elements of a sequence as the sequence converges. Suppose x_n is a convergent sequence and define d_n = |x_n - L|.

Theorem 144

$lim\ x_n = L \Leftrightarrow \ lim\ d_n = 0$

We state a result here. The converse of the following is quite important and will be proven later. The basic idea of the following is that elements of a convergent sequence eventually get close together. This might seem obvious to you. The converse, that if the elements of a sequence get arbitrarily close together then the sequence is convergent may also seem obvious but takes a bit more machinery to prove.

Theorem 152

Suppose that $lim\ x_n = L$ . Then, for any $\epsilon \gt 0$ , there is a number $N_{\epsilon}$ large enough so that, for all n and m greater than $N_{\epsilon}$ , $\vert x_n - x_m \vert < \epsilon$ .

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William Thistleton
11/10/1998