Definition.

A sequence is a function a:A→R where either A=Z+:={1,2,3,…} or A=N:={0,1,2,3,…} (the natural numbers) or perhaps A is any countable set.1 Thus, either

or

where in both cases an∈R. We refer to an as the n-th element or entry of the sequence.

Remarks.

1.

Notice that a sequence differs from a set in that a sequence has an order, whereas a set is simply a collection of elements. Technically, there is no first element in a set, but there is in a sequence.

2.

This is not the most general definition of the word “sequence,” but it is easily general enough for our purposes. It contains all the key ideas to discuss sequences. In particular, we will not discuss finite sequences.

3.

Notice that for our purposes, sequences begin either with n=0 or with n=1. Beginning with a zeroth element is less common, but it occurs sometimes when a zeroth element makes sense. Among other places, this happens in computer science. Still, unless there is some specific reason to do otherwise, let A=Z+.

Example 1.1.1.

Suppose that an:=1/n (i. e., an is defined equal to 1/n), so that

Notice that this sequence must start with n=1 since a0 is undefinable using the given formula. Also notice that an decreases toward zero as n increases toward infinity, even though an≠0 for any n∈Z+. In symbols, that is an↘0 as n↗∞.

All of this may seem clear intuitively, but as we will see, intuition is not always so clear. So again an exact mathematical definition is needed.

Definition.

A sequence {an}⊂R converges to a limit L∈R iff given ε>0, ∃N∈Z+ such that |an−L|<ε whenever n>N. Written with a minimal amount of symbols, this definition is “a sequence {an} of real numbers converges to a limit L in the reals if and only if given ε>0, there exists N in the positive integers such that |an−L|<ε whenever n>N.” The sequence {an} diverges iff it does not converge.

Notation.

The notation to indicate that a sequence {an} converges to a limit L is

or more briefly

Remark.

In this definition, ε can be any positive real number, but it is generally thought of as a small positive real number. The definition works because ε can be an arbitrary small positive real number. A schematic diagram of the relationship between ε, L and the elements of the sequence ak is shown in Figure 1.2.

Figure 1.2 A sequence approaching a limit L on the real line. The early elements of the sequence through aN can be outside the interval (L−ε,L+ε), but starting with aN+1, all of the elements must lie inside the interval. In particular, when n>N, then an must lie in the interval. Where might aN+5 be on this number line? If ε is made smaller, then likely N will have to be made larger to keep the elements of the sequence from aN+1 onward in the new smaller interval.

Example 1.1.1.

Returning to our first example, suppose again that an:=1/n. Does the sequence {an} converge, and if so, to where? For a given ε, what value of N guarantees that the convergence definition is satisfied?

Remarks.

1.

Notice that the value of N increases as ε decreases; this is almost always the case. Expect ε to be in the denominator in the expression for N.

2.

Notice the direction of the double arrow above. Although we started on the left working on |an−L|, we need to keep in mind that this is the conclusion to be reached provided that n is large enough. So a backward implication (a backward double arrow) is what is needed. Indeed, the symbol “⟸” is perhaps best read as “whenever”. Thus, the implication above becomes |an−L|<ε whenever n>1/ε.