1549901369-Elements_of_Real_Analysis__Denlinger_

2.2 Algebra of Limits 63

Theorem 2.2.3 A constant sequence {xn} = {c} converges (to c).

Proof. Exercise 3. •

Definition 2.2.4 A sequence {xn} is said to be eventually constant if :Jc E

JR and 3 no E N 3 'in 2 no, Xn = c.

Theorem 2.2.5 An eventually constant sequence converges (to that constant).

Proof. Exercise 4. •

Theorem 2.2.6 (Fundamental Limit) lim ~ = 0.
n->oo n

1. Proof. Let c: > 0. Then - > 0. By the Arch1medean property, 3 n 0 E N 3
   c:
   1
   no>-. Then
   c:

1

n 2 no::::} n > -

c:

. 1
Therefore, l!m - = 0. •
n--+oo n

Theorem 2.2.7 (Uniqueness of Limits) A sequence cannot converge to
more than one real number.

Proof. Suppose {xn} is a sequence, with lim Xn =Land lim Xn = M.
n--+oo n--+oo
We want to prove that L = M. We shall use the "forcing principle" [Theorem
1.5.9 (d)J. Let c: > 0.

. c:
Smee Xn -+ L, 3 n1 E N 3 n 2 n1 ::::} lxn - LI < 2"
. c:
Smee Xn-+ M, 3n2 EN 3 n 2 n2::::} lxn - Ml< 2·