9. Interpret a decimal expansion 0.ala2 ... as Prove that

(a) 0.5 = 0.4999 ... and

(b) 1 = 0.999 ...

[!ill. This exercise was used in Section 1.4.

(a) Suppose that 0 ::; Y < IlIOn for some integer n ~ O. Prove that there is an integer 0 ::; W ::; 9 such that W W 1 Ion+! ::; Y < IOn+1 + IOn+I'

(b) Prove that given x E [0,1) there exist integers 0 ::; Xk ::; 9 such that for all nEN,

~ Xk ~ Xk 1

~ 10k ::; X < ~ 10k + IOn'

k=l k=l

(C) Prove that given x E [0,1) there exist integers 0 ::; Xk ::; 9, kEN, such that n

. '" Xk x= n~~ 10k'