1549901369-Elements_of_Real_Analysis__Denlinger_

5.6 *Exponentials, Powers, and Logarithms 279

any rational number greater than x, then \:/n E N, rn :::; x < r, so arn < ar.
Hence { arn} is bounded above. Thus, by the monotone convergence theorem,
{ arn} converges. •

Lemma 5.6.4 Let a ~ 1 and x E R If {rn} and { sn} are monotone increasing
sequences of rational numbers converging to x, then lim arn = lim a^5 n.
n--+oo n--+oo
Proof. Suppose a, x, {rn} and {sn} satisfy the hypotheses. Since the
theorem is trivially true when a= 1, we shall assume a> 1. By Lemma 5.6.3,
::3 L = lim arn and ::3M= lim a^5 n. We shall prove L = M.
n--+oo n--+oo
Since {rn} and {sn} are monotone increasing, and the function f(r) = ar
is strictly increasing on Q, { arn} and { a^5 n} are monotone increasing, so

L = sup{arn : n EN} and M = sup{a^5 n : n EN}.

Define a new sequence {Sn} by Sn = Sn - ~- Then {Sn} is a strictly
increasing sequence of rational numbers with

lim a^8 n = lim asn-!; = lim as: = M = M. (See Example 2.3.9.)

n--+oo n---too n--+oo an 1

Let c: > 0. Then ::3 n 1 EN 3

M - c: < a^8 n1 :::; M. (14)

Now {Sn} is a strictly increasing sequence converging to x, so

Since { r n} is a monotone increasing sequence converging to x, ::3 n2 E N 3

and, since the function f(r) = ar is strictly increasing on Q,

(15)

Putting together (14) and (15) we have, since limits preserve inequalities,

n->oo

M - L:::; c:.

Since this holds for all c > 0, the forcing principle implies

M - L:::; O; i.e.,

M:::; L.