1549901369-Elements_of_Real_Analysis__Denlinger_

2.3 Inequalities and Limits 75

Now~ -t 0. Therefore, by the second squeeze principle, with an=

2

n +^3

n 3n - 7 '

2 3 2n + 3 2

L = -

3

, bn =-,and no= 22, we have proved that ---t -. D

n 3n - 7 3

Example 2.3.4 Use the second squeeze principle to prove that

. (3n

2

hm - 4n)

n-+oo n^2 + 5 = 3. (See Example 2.1.8.)

Solution: 'Vn EN,

I

3n

2

• 4n _ 31 = I (3n

2

• 4n) - 3 (n

2

+ 5) I

n^2 + 5 n^2 + 5

= 1-4n- 151

n^2 + 5

4n+ 15

n^2 + 5

4n+n

< -- 2 - ifn :2:: 15

n

5n 5

n^2 n

That is, n :2:: 15 ~ I 3n

2

• 4
  n - 31 <
  n^2 +5 n

5

-t 0. Therefore, by the second

.. 1 3n2 - 4n 3 D
squeeze prmc1p e, n 2 +
5
-t.

MORE APPLICATIONS OF THE SQUEEZE PRINCIPLE

Theorem 2.3.5 Let A s;;; IR.

(a) If u =inf A, then 3 sequence {an} of elements of A such that an-tu.

(b) If u =sup A , then 3 sequence {an} of elements of A such that an-tu.

1
Proof. (a) Suppose u =inf A. Let n EN. Then u+ - is not a lower bound
n
1
for A , so :Jan EA 3 an< u + - (see Theorem 1.6.7, c:-criterion for infimum).
n