166 Chapter 3 • Topology of the Real Number System
E ach base-b decimal represents a unique real number. As in Section 2.5,
Vn E N, we define
n d ·
D:i = K.d 1 d2 · · · dn (base b) = K +Lb;
i=land observe that {Dn} is a bounded monotone sequence. By the monotone
convergence theorem, this sequence has a limit, which we sha ll call D. It is in
this sense that we say
K.d1d2 · · · dndn+l · · · (base b) = D. 0
We can modify the proofs of Theorems 2.5.5 and 2.5.7 , and Example 2 .5. 6
to yield:.
Theorem 3.4.6 Given any natural number b > 1,
(a) Every base-b decimal represents a unique real number.(b) Every real number can be represented by a base-b decimal.( c) Some real numbers can be represented by two base-b decimals, one ending
in all O's, and another ending in all (b - 1) 's.Proof. (Omitted) •For example, in base-3,
2.10022222 ... (base 3) = 2.1010000 ... (base 3) = 2+ i + 217 = 2~~·Definition 3.4. 7 A terminating decimal in base-b is one ending in all O's.
(All others are called nonterminating.)
Theorem 3.4.8 L et b be a natural number greater than 1. There is a 1-1 cor-
respondence between JR and all nonterminating base-b decimals. (That is, every
real number has a unique nonterminating decimal in base-b.)
Proof. (Omitted) •Examples 3.4.9 In base-3, only 0, 1, and 2 are used as "digits." We h ave
(a) 0.1 (base 3) = i;
(b) 0. 12 (base 3) = i + ~ = ~;
(c) 0.021 (base 3) =~+~+A= 277 ;