1549901369-Elements_of_Real_Analysis__Denlinger_

108 Chapter 2 • Sequences

The following lemma is simply a technical observation that frequently

proves useful in what is to come.

Lemma 2.6.3 If { nk} is a strictly increasing sequence of natural numbers,

then Vk E N, nk 2 k.

Proof. Exercise 1. •

In the following definition and subsequent results, we develop some useful

t erminology.

Definition 2.6.4 (a) A sequence {xn} is said to be eventually in a set A , if

:3 n 0 E N 3 Vn 2 no, Xn E A.

(b) A sequence { Xn} is said to be frequently^11 in a set A, if Vn 0 E N,

:3 n 2 no 3 Xn E A.

In wor.ds, a sequence is "eventually in" a set A if some tail of the sequence

is in A; a sequence is "frequently in" A if every tail of the sequence contains

a member of A.

Example 2.6.5 The sequence { 2 + ~} is eventually in the interval (1.99, 2.01).

[That interval contains t he 101-tail of the sequence.] The sequence

{ ( -1 r ( 2 + ~) } is frequently in , but is not eventually in , the interval

(1.99, 2.01). [No tail of the sequence is in that interval, but every tail does

contain some member of the interval.]

Lemma 2.6.6 (a) A sequence {xn} is eventually in a set A¢:> A contains all

but a finite number of terms of { Xn}; that is, A contains Xn for all but finitely

many n EN.

(b) A sequence {xn} is frequently in a set A ¢=> A contains infinitely.

many terms of { Xn}; that is, A contains Xn for infinitely many n E N.

Proof. Exercise 2. •

We put this new terminology to work in the following theorem.

Theorem 2.6.7 Let {xn} be a sequence and let L be a real number. Then

(a) Xn converges to L ¢:>Ve> 0, {xn} is eventually in (L - e, L + e).

(b) {xn} has a subsequence converging to L ¢=>Ve> 0, {xn} is frequently in

(L-e,L+e).

11. Although "frequently" is in common usage, " infinitely often" might be a better way to
    describe this situation.

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