1549901369-Elements_of_Real_Analysis__Denlinger_

280 Chapter 5 • Continuous Tunctions

Reversing the roles of { r n} and {Sn} will allow us to prove L :::; M. There-

fore, L = M. •

Because of Lemmas 5.6.2-5.6.4 we can make the following definition:

Definition 5.6.5 (Exponential Functions ax, where a > 0)

Let a ;:::: 1. Then '<Ix E JR, we define ax = lim arn , where {rn} is any

n--->oo

monotone increasing sequence of rational numbers converging to x.

1

If 0 <a< 1, then a-^1 > 1, so '<Ix E JR, we define ax = (a-l )x.

Note that these two definitions are consistent since, when 0 < a < 1 and

{rn} is any monotone increasing sequence of rational numbers converging to x ,

n--->OO lim arn = n--->OO lim ( a -~) Tn lim / a-1 )Tn ( a !l) X = ax. Thus, Va ;::::^0 and

n--->oo

'<Ix E JR, ax= lim arn.

n--->oo

Remark: When xis a rational number, we now have two definitions of ar, the

definition just given and the one given in Exercise 5.5.15. To see that these two

definitions agree when x is a rational number, just use the constant sequence

{ Xn} = { x} in Definition 5.6.5, and apply Lemma 5.6.4.

Theorem 5.6.6 Let a> 1. Then the exponential function f(x) =ax defined

in Definition 5. 6. 5 is a strictly increasing positive-valued function with domain

JR whose range has no upper bound.

Proof. Let a > 1. Consider the exponential f ( x) = ax defined in Definition
5.6.5.
(1) 7J(f) = R
(2) Let x < y in R Since the rationals are dense in JR, 3 rational number
q such that x < q < y. Let { r n} and {Sn} be monotone increasing sequences of
rational numbers such that r n --t x and Sn --t y. Since the terms of {Sn} must
eventually be greater than q, we may assume, without loss of generality, that
'<In EN,

r n :S X < q < Sn :S Y.

Then, by Lemma 5.6.1,

Thus, since limits preserve inequalities, and since { a^8 n} is monotone increasing,

n--->oo n--->oo

Thus, ax < aY; that is, f is strictly increasing on R