92 Chapter 2 • SequencesWe can use sequences to make this meaning precise. To keep it simple, we shall
consider only the case when D 2: 0. The case D < 0 is handled similarly.
First, notice that by definition the decimal ( 4) has infinitely many "digits."
We are accustomed to thinking of some decimals, such as ~ = 0.5 as "termi-
nating;" i.e., having only finitely many digits. Let us agree that such decimals
really have infinitely many digits, by making all remaining digits zeros.
For each n E N, we define then-place truncation of D to be the terminating
decimal
Dn = K.d1 d2d3 · · · dn
by which we mean (the rational number)
D -K d1 ~ ... dn
n - + 10 + 100 + + ion.
Now, it is clear that {Dn} is a monotone increasing sequence, bounded
above, since
D1 ::; D2 ::; · · · Dn ::; · · · ::; K + l.
Therefore, by the monotone convergence theorem, there is a (unique) real
number x E JR 3 Dn --+ x. It is this real number, x, that is "represented" by the
decimal expansion (4). We summarize these results in the following theorem.
*Theorem 2 .5.5 Every decimal expansion D = K.d 1 d 2 d 3 · · · dndn+l · · · de-
fined above represents a unique nonnegative real number; namely, x = sup{Dn :
n EN}, or x = lim Dn, where Dn is as defined above.
n->oo
Theorem 2.5.5 establishes a one-way relationship between decimals and
nonnegative real numbers,
decimals --+ nonnegative real numbers.
We have not yet proved that the relationship goes the other way as well:
that to each nonnegative real number there corresponds a decimal expansion
of the form (4). We now turn our attention to that concern.
Let x 2: 0. (We consider here only the case x 2: 0. To represent x < 0 as a
decimal, represent !xi as a decimal and prefix a "-" .) In Chapter 1 (Theorem
1.5.3) we showed that 3 unique nonnegative integer K 3
K ::; x < K + l. Then
0::; x - K < 1, so
0 ::; lOx - lOK < 10.Hence, 3 unique integer d 1 E {1, 2, · · · , 9} such thatd1 ::; lOx - lOK < d 1 + 1
0 ::; lOx - lOK - d 1 < 10 ::; 10 [ x - ( K + ~~)] < 1