1549901369-Elements_of_Real_Analysis__Denlinger_

2.3 Inequalities and Limits 79

LIMITS PRESERVE INEQUALITIES

In our future work we shall have many occasions to use the following two

theorems.

Theorem 2.3.12 (Limits Preserve Inequalities, I)

(a) If {an} converges and Vn EN, an::::; K, then lim an::::; K.

n-+oo

(b) If {an} converges and Vn EN, an 2". K, then n-+oo lim an 2". K.

Proof. (a) Suppose {an} converges and Vn EN, an::::; K. Say an---+ L. We

must prove that L ::::; K. For contradiction, suppose L > K. Let c: = L - K.

Then c: > 0. Since an---+ L, ::J no EN 3

n 2". no :=? Ian - LI < c:

Therefore, L ::::; K.

(b) Exercise 12. •

:=? -c: <an -L < c:

:=?L-c:<an<L+c:

==> L - (L - K) < an

:=? K <an; contradiction!

Theorem 2.3.13 (Limits Preserve Inequalities, II) If {an} and {bn} are

convergent sequences, and Vn EN, an ::::; bn, then n--+oo lim an ::::; n--+oo lim bn.

Proof. Suppose {an} and {bn} are convergent sequences, and Vn E N,
an ::::; bn. Define the sequence {en} by Cn = bn - an. Then, by the algebra of
limits, {en} converges and
lim Cn = lim (bn - an) = lim bn - lim an.
On the o~~; hand,nv;.ooE N, :!;, 2". 0~;

00
by Th~-::r':m 2.3.12, lim Cn 2". 0.
n-+oo
That is, lim bn - lim an 2". O; i.e., lim an ::::; lim bn. •
n--+oo n-+oo n--+oo n--+oo

Theorem 2.3.14 (Partial Converse of 2.3.13) If {an} and {bn} are con-

vergent sequences such that lim an < lim bn, then ::J no E N 3 n 2". no ==> an <

n--+oo n-+oo

bn.

Proof. Exercise 16. •