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
- 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·