Sequences and limits

A sequence {xn} 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 {kn} is a strictly increasing sequence of positive integers, the sequence {xkn} is called a subsequence of {xn}. The definition implies that {xn} is a subsequence of itself.

A sequence {xn} 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 {xn} 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 {xn} converges to x, the sequence {xn − 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 {xn} converges to x is thus the same thing as saying that the sequence of points {xn} converges to the point x, or that the distance between the point xn and the point x tends to zero as n → ∞.