1 - Introduction

Summary

Sequences and limits

A sequence {x_n} is a collection of objects occurring in order; thus there is a first member x₁, a second member x₂, and so on indefinitely. For every positive integer k, there is a corresponding kth member of the sequence. The members of such a sequence need not be all different. We can have a sequence all of whose members are the same; such a sequence is called a constant sequence.

If {k_n} is a strictly increasing sequence of positive integers, the sequence {x_kn} is called a subsequence of {x_n}. The definition implies that {x_n} is a subsequence of itself.

A sequence {x_n} of real numbers is said to converge to the limit x if, for every positive value of ∈, all but a finite number of members of the sequence lie between x – ∈ and x + ∈. If a sequence {x_n} of real numbers converges, every subsequence converges to the same limit. A sequence of real numbers which converges to zero is called a null-sequence. Thus if {x_n} converges to x, the sequence {x_n − x) is a null-sequence.

It is often convenient to represent real numbers by points on a line, and to speak of the point of abscissa x simply as the point x. The distance between the points x and y is |x − y|. To say that the sequence of real numbers {x_n} converges to x is thus the same thing as saying that the sequence of points {x_n} converges to the point x, or that the distance between the point x_n and the point x tends to zero as n → ∞.