1549901369-Elements_of_Real_Analysis__Denlinger_

2.9 *Upper and Lower Limits 133

(d) Vn E N, Wn = - oo, so lim Wn = - oo. Also, Vn E N, Wn = -n, so

n-+oo

n-+oo lim Wn = - oo. D

Remarks 2.9.5 (a) Since lim Xn = lim (sup{xk: k ~ n} ), lim Xn is of-

n-+oo n-+oo n--+oo

ten called the lim sup (or limit superior) of {xn}· Similarly, lim Xn =

n-+oo

lim (inf { x k : k ~ n}), so lim Xn is often called the lim inf (or limit infe-

n--+oo n-+oo

rior) of { Xn}. These quantities are often denoted lim sup Xn and lim inf Xn,

n-+oo n-+oo

respectively.

(b) Since { Xn} is monotone decreasing, the monotone convergence theorem

(2.5.3) tells us that

lim Xn = inf{xn: n EN}

n-+oo

= inf {sup{xk : k ~ n}: n EN}.

Similarly, { Xn} is monotone increasing, so

lim Xn = sup{xn: n EN}

n-+oo

= sup {inf{xk: k ~ n}: n EN}.

Theorem 2.9.6 (Elementary Properties of Upper and Lower Limits)

(a)

n-+oo

lim Xn :::; lim Xn.

n-+oo

(b) If { Xn} is bounded above by B, then lim Xn :::; B.

n-+oo

(c) If {xn} is bounded below by A, then lim Xn ~A.

n-+oo

(d) If {xnk} is any subsequence of {xn}, then

lim Xn :::; lim Xnk :::; lim Xnk :::; lim Xn.

n--+oo k--+oo k--+oo n-+oo

Proof.

(a) Vm,n EN, Xn:::; Xn+m :::; Xn+m :::; Xm· Thus, sup{xn : n EN} :::;

inf{xn : n EN}. (See Exercise 1.6-B.3.) That is, n-+oo lim -Xn :::; m--+oo lim Xm (by the

monotone convergence theorem, 2.5.3). Therefore, lim Xn :::; lim Xn·
n-+oo n-+oo
(b) Suppose that Vn EN, Xn :::; B. Then Bis an upper bound for every
n-tail of {xn}, so Xn = sup{xk: k ~ n}:::; B. Thus, lim n-+oo Xn:::; B. (Why?) That
is, lim Xn :::; B.
n-+oo
( c) Exercise 2.
(d) Suppose {xnk} is a subsequence of {xn}· Let m EN. Then, Vk E N,
nk ~ k so {xnk : k ~ m} ~ {xn: n ~ m}. Thus, Xn= ~ Xm and Xn,,, :::; Xm·