Calculus: Analytic Geometry and Calculus, with Vectors

5.4 Theorems about continuous and differentiable functions^321

18 Prove that if x is a number, then there exist an integer N and a sequence
d1, d2, d3, of digits (a digit being one of the integers 0, 1, 2, , 9) such

that

dl d2 d

(1) N+10+102+

-10-1+1Onsx

d, d2 do-1 do + 1

< N +10 + 102+ ... + 10n-1 + 10^

for each n = 1, 2, 3,. Solution: Let N = [x], so that N is the greatest

integer in x, and let 0 = x - N. Then 0 5 0 < I and the required result will

follow if we prove that there exist digits d1, d2, da, such that

(2) 10 + 02 + ...

for each n = 1, 2, 3,

(3)

do-1 do

10-1 10,

<10+2 0+

d2 10n-1 +do-1 do + 1

10,

. To prove (2), it is sufficient to prove that

n

05 9-1o- 022- ... 0n<iOn-

While it is of interest to take time to use (2) to determine what dl, d2, da,. ..

must be if they exist, we save time by definingintegers d,, d2, da, by the

formulas

(4)

(5)

(6)

and, in general, for

(7)

Since 0 <= 0 <

over,

d, = [100]

d2 = [102(0 l1

OIJ

d3 =[10a(6-10 1021

each n=1,2,3,.

pn)J

1, we find that 0 5 106 < 10 and hence that d, is adigit. More-

d, 5 100 < d, + l

056-10<10'

so (3) holds when n = 1. Multiplying (9) by 102 gives

(10) 05102(0-10)<10.