1549901369-Elements_of_Real_Analysis__Denlinger_

7 4 Chapter 2 • Sequences

Let no= max{n1,n2}. Then

n 2: no ==? n 2: n1 and n 2: n2

=} Ian - LI < € and lcn - LI < €

=} -€ < an - L < € and - € < Cn - L < €

=} L - € < an < L + € and L - € < Cn < L + €

=} L - € < an and Cn < L + €

=} L - € < an :::; bn :::; Cn < L + €

=* L - c < bn < L + c

=} lbn - LI < €.

Therefore, bn ----> L. •

Corollary 2.3. 2 (The Second "Squeeze" Principle) If {an} and {bn} are
sequences such that bn ----> 0 and 3 no E N 3 n 2: no =} Ian - LI :::; bn, then
an----> L.

Proof. Exercise 1. •

USING THE SQUEEZE PRINCIPLES TO PROVE

CONVERGENCE

The second squeeze principle provides an easy way to convert the "scratch-
work" done in Section 2.1 (as in Part (c) of Examples 2.1.5 and 2.1.7) into
proofs, without having to rewrite the work into a proof like we did in that
section (as in Examples 2.1.6 and 2.1.8). The following two examples illustrat e
this procedure.

Example 2.3.3 Use the second squeeze principle to prove t hat

(
lim --2n + 3) = -.^2 (See Example 2.1.6.)
n->oo 3n - 7 3

S o (^1) ut1on... w vn E 1'l, i;,,_r I --2n +^3 -^2 - I - I - 3(2n + 3) ( -^2 (3n ) - 7) I
3n - 7 3 3 3n - 7
= I 6n + 9 - 6n + 141
3(3n - 7)
23
l9n - 211
23.
9n - 21 if n 2: 3
23
< --ifn> 22
9n-n -
24 3
< - = - if n > 22.
8n n -