1549901369-Elements_of_Real_Analysis__Denlinger_

132 Chapter 2 • Sequences

Definition 2.9.2 (Lower Limit) Suppose {xn} is any sequence ofreal num-

bers. Case 1: If { Xn} is bounded below we define, \:/n E N,

Xn = inf{xk: k 2: n}.

That is, Xn is the infimum of the n-tai1^18 of {xn}· Since {xn} is bounded

below, each Xn is a real number. Note that

We define the lower limit of { Xn} to be

lim Xn = lim Xn·

n-->oo n--+oo-

Since { Xn} is monotone increasing, we know that lim Xn is either a real

n-->oo

number or +oo.

Case 2: If {xn} is not bounded below then \:/n E N , Xn

define

lim Xn = lim (-oo) = -oo.

n-->oo n-->oo

-oo and we

Remark 2.9.3 Although not every bounded sequence converges, every

bounded sequence has both an upper limit and a lower limit, which are unique,

finite real numbers. Even unbounded sequences have upper and lower limits; if

a sequence is not bounded above, its upper limit is +oo, and if a sequence is

not bounded below, its lower limit is -oo.

Examples 2.9.4 Find t he upper and lower limits of each of the following se-

quences:

(a) {xn}={l , 0, 1, 0, 1,0,1,-··}

(b) {Yn} = {1, ~, 3 , i,5, i, 7, ~,-· ·}

(c) {zn} = {1, -~, ~, -i, t> -i, ~' -~, · · ·}

(d) {wn} = {-n}.

Solutions:

(a) \:/n EN, Xn = 0, so lim Xn = lim Xn = lim 0 = 0.

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

Similarly, lim Xn = lim Xn = lim 1 = 1.

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

(b) \:/n EN, Yn = 0, so lim Yn = lim Yn = lim 0 = 0.

• n--+oo n--+oo - n--+oo
  Also, n--+oo lim Yn = n--+oo lim Yn = n--+oa lim ( +oo) = +oo.

(c) Note that {z __!!. } = {-1. 2> _l. 2> _l. 4> _l. 4' _l. 6' 6' _l. _l. 8' 8' _l. ···}·th> us > 11·m - z n --

n-->oo

0. Similarly, {zn} = {1, ~, ~, t> t> ~' ~,···};thus, lim Zn= 0.
   n-->oo