1549901369-Elements_of_Real_Analysis__Denlinger_

104 Chapter 2 • Sequences

{an} and {bn} of left and right endpoints of the intervals In. Since the intervals

are nested, we have Vn EN,

Thus, the sequence {an} is monotone increasing and bounded above by b 1.

Likewise, the sequence {bn} is monotone decreasing and bounded below by a 1.

By the monotone convergence theorem,

3a = n--+oo lim an = sup{ an: n EN}, and 3b = n--+oo lim bn = inf{bn: n EN}.

Now, Vn E N, an ::; bn. Thus, by Theorem 2.3.12, a ::; b. We shall prove

00

that n I n = [a, b].

n=l

Let x E [a, b]. Then Vn EN, an::; a::; x::; b::; bn, so x E In. Thus, Vn EN,

[a, b] s:;; In. Therefore,

00

[a,b] s:;; n In. (9)

n=l

We now show that the set containment goes the other way also. Suppose

00

y E n In. Then Vn E N, an ::; y ::; bn. Then, by Theorem 2.3.12 (limits

n=l

preserve inequalities), lim an ::; y::; lim bn. That is, a::; y::; b. Therefore,

n-+oo n-+oo

00

n Ins:;; [a,b]. (10)

n=l

Putting (9) and (10) together, we have

00

n In= [a,b]. (11)

n=l

Since a ::; b, this interval is nonempty.

(b) Suppose further, that lim (bn-an) = 0. Then, by the algebra of limits,
n--->oo
lim bn - lim an = 0. But lim bn = b and lim an = a. Thus, b - a = O;
n-+oo n-+oo n-+oo n--+oo
that is , b = a.
00
Therefore, by (11), n In= {a}. •
n=l

EXERCISE SET 2.5

l. A sequence {xn} is said to be eventually monotone (increasing or

decreasing) if it has a tail that is monotone (increasing or decreasing).

Restate Theorem 2.5.3 for eventually monotone sequences, and prove ei-

ther (a) or (b) of the resulting theorem.