52 Chapter 2 • Sequences
We can also plot the terms of the sequence as points on a number line, as
in Figure 2.1 (b), and notice that the successive terms of the sequence "cluster"
around the limit, 2. We will find this notion of clustering to be a useful one
later in the chapter.
We are now ready for the official definition of convergence and limit.Definition 2.1.4 Let {xn} be a sequence and L be a real number. Then
n->oo lim Xn = L if Vt> 0, 3no EN^3 \In EN, n^2 no=} lxn - LI< t.^2
If lim Xn = L, we say that {xn} converges to L; and write
n->OOIf there is no real number to which { Xn} converges, we say that { Xn}
diverges.
USING DEFINITION 2.1.4 TO PROVE THAT lim Xn = L
n->oo
Using Definition 2.1.4 will require a mental change of gears. Ignoring the
quantifiers for the moment, notice that implementing the definition will require
us to prove that one inequality implies another. Specifically, we must show that
the inequality n 2 no implies the inequality lxn - LI < t. When we throw in
the quantifiers Vt > 0, 3 n 0 E N, and \In E N, it can look pretty confusing!
The secret to avoiding confusion is to understand what the definition means.
Intuitively, it means we must show that corresponding to an arbitrary positive
real number t, after a certain number no of terms, all the remaining terms of
the sequence will be within a distance t from L.
Verbal paraphrases of Definition 2.1.4:^3n->oo lim Xn = L means:
- Xn can be made arbitrarily close to L by making n sufficiently large.
- lxn - LI can be made arbitrarily small by making n sufficiently large.
' • For every positive t there is some n 0 such that lxn - LI < t whenever
n 2 no.
Notice the important role played by inequalities in Definition 2.1.4. Why
inequalities? The simplest explanation for this is that analysis must deal with
infinity; both the infinitely large and the infinitely small. Since no real number
- In practice, we usually leave out the third quantifier (Vn E N) in the interest of simplicity.
It is understood to be present even when not written. - A lthough Definition 2.1.4 is officially correct, and should be memorized, it is equally im-
portant (for the sake of understanding) to be able to paraphrase it in words.